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A   TREATISE    ON 


GEAR  WHEELS, 


BY  GEORGE  B.  GRANT. 
h 


51 TD 


PUBLISHED  BY 

GEORGE    B.    GRANT, 

LEXINGTON,  MASS. 
PHILADELPHIA,  PA. 


SIXTH    EDITION 


George  B.  Grant, 

86  Seneca  St. 
Cleveland,  -  Ohio. 


PRICK,     clotln     and.     gilt,     $1.OO      post-paid.   / 
paper    covers,  .60  " 


AQE.NTS     WANTKD. 

Any  active  and  intelligent  man  can  make  money  selling  this  book.     I   allow  a  liberal  dis- 
count and  take  back  unsold  copies.     Write  for  circular  of  "  terms  to  agents." 


Contributed    to    ttx© 

AMERICAN     MACHINIST, 
Kor  1890. 


COPIED  from  the  American  Machinist,  and  published  in  full  by  the  "  English  Mechanic  and 
World  of  Science,"  and  by  the  "  Mechanical  World." 

ADOPTED  and  used  as  a  text  or  reference  book  by  Michigan  and  Cornell  Universities,  Rose 
Polytechnic  Institute,  and  by  many  schools  of  mechanics  and  drafting.  - 


Copyright    1893. 
By    GEORGE     B.     GRANT. 


A    Working    Course   of  Study. 

It  is  not  necessary  that  the  student,  especially  if  he  is  a  workman,  should 
learn  all  that  is  taught  in  this  book,  for  it  contains  much  that  is  not  only  difficult 
but  also  of  minor  practical  importance. 

The  beginner  is  therefore  advised  to  master  only  the  following  sections  : 

i,  2,  7  to  15,  22,  25,  31,  32  of  the  general  theory; 

-35  to  47  of  the  spur  gear  ; 

53  to  64  of  the  involute  tooth  ; 

76,  77,  80  to  83,  89  of  the  cycloidal  tooth  ; 

91,  95,  97  of  the  pin  tooth  ; 

98,  99,  in,  113  to  119  of  spiral  and  worm  gears; 

154,  157,  158,  161  to  169  of  the  bevel  gear. 

These  include  not  half  of  the  whole  matter,  but,  knowing  this  much  well, 
the  student  has  a  good  outline  knowledge  of  the  whole,  and  he  can  then  take 
the  balance  at  leisure. 


A    TREATISE    ON 


GEAR      WHKELS 


1.    THEORY    OK    TOOTH    ACTION. 


1 . — INTRODUCTORY. 


The  present  object  is  practical,  to  reach 
and  interest  the  man  that  makes  the  thing 
written  of ;  the  machinist  or  the  millwright 
that  makes  the  gear  wheel,  or  the  drafts- 
man or  foreman  that  directs  the  work,  and 
to  teach  him  not  only  how  to  make  it,  but 
what  it  is  that  he  makes. 

To  most  mechanics  a  gear  is  a  gear. 

"  A  yellow  primrose  by  the  shore, 
A  yellow  primrose  was,  to  him, 
And  it  was  nothing  more ; " 

and,  in  fact,  the  gear  is  often  a  gear  and 
nothing  more,  sometimes  barely  that. 

But,  if  the  mechanic  will  look  beyond  the 
tips  of  his  fingers,  he  will  find  that  it  can 


be  something  more  ;  that  it  is  one  of  the 
most  interesting  objects  in  the  field  of  scien- 
tific research,  and  not  the  simplest  one  ;  that 
it  has  received  the  attention  of  many  cele- 
brated mathematicians  and  engineers  ;  and 
that  the  study  of  its  features  will  not  only 
add  to  his  practical  knowledge,  but  also  to 
his  entertainment.  There  is  an  element  in 
mathematics,  and  in  its  near  relative,  theoreti- 
cal mechanics,  that  possesses  an  educating 
and  disciplining  value  beyond  any  capacity 
for  earning  present  money.  The  thinking, 
inquisitive  student  of  the  day  is  the  success- 
ful engineer  or  manufacturer  of  the  future. 


2. — METHOD. 


The  method  will  be  fitted  to  the  object,  and 
will  be  as  simple  and  direct  as  possible.  It  is 
not  possible  to  treat  all  the  items  in  simple 
every-day  fashion,  by  plain  graphical  or  arith- 
metical methods,  but  where  there  is  a  choice 
the  path  that  is  the  plainest  to  the  average 
intelligent  and  educated  mechanic  will  be 
chosen. 

A  thousand  pages  could  be  filled  with  the 
subject  and  not  exhaust  anything  but  the 
reader  thereof,  but  what  is  written  should 
receive  and  deserve  attention,  and  must  be 
condensed  within  such  reasonable  limits,  that 
it  shall  not  call  for  more  time  and  labor  than 


its  limited  application  will  warrant.  Demon- 
strations and  controversies  will  be  avoided, 
and  the  matter  will  be  confined  as  far  as  is 
possible  to  plain  statements  of  facts,  with 
illustrations.  The  simplest  diagram  is  often 
a  better  teacher  than  a  page  of  description. 
First,  we  shall  study  the  odontoid  or  pure 
tooth  curve  as  applied  to  spur  gears,  then 
we  shall  consider  the  involute,  cycloid,  and 
pin  tooth,  special  forms  in  which  it  is  found 
in  practice ;  then  the  modifications  of  the 
spur  gear,  known  as  the  spiral  gear,  and  the 
elliptic  gear ;  then  the  bevel  gear,  and  lastly 
the  skew  bevel  gear. 


Literature. 


3. — PARTICULARLY  IMPORTANT. 


Begin  at  the  beginning. 

The  natural  tendency  is  too  often  to  skip 
first  principles,  and  begin  with  more  ad- 
vanced and  interesting  matter,  and  the  result 
is  a  trashy  knowledge  that  stands  on  no 
foundation  and  is  soon  lost.  When  a  fact 
is  learned  by  rote  it  may  be  remembered, 
but  when  it  follows  naturally  upon  some 
simple  principle  it  cannot  be  forgotten. 

Therefore  the  student  is  urged  to  begin 
with  and  pay  close  attention  to  the  odontoid 
or  pure  tooth  curve,  before  going  on  to  its 


special  applications,  for  the  apparently  dry 
and  trivial  matter  relating  to  it  is  really  the 
foundation  of  the  whole  subject. 

The  usual  course  is  to  begin  at  once  with 
the  cycloidal  tooth,  to  hurry  over  the  in- 
volute tooth,  and  then,  if  there  is  room,  it 
is  stated  that  such  curves  are  particular 
forms  of  some  confused  and  indefinite  general 
curve.  Our  course  will  be  to  study  the  unde- 
fined tooth  curve  first,  and  then  take  up 
its  particular  cases. 


4. — LITERATURE. 


It  is  impracticable  to  acknowledge  all  the 
sources  from  which  information  has  been 
drawn,  but  it  is  in  order  to  briefly  mention 
the  principal  works  devoted  to  the  subject. 

Professor  Herrmann's  section  of  Professor 
Weisbach's  "Mechanics  of  Engineering  and 
Machinery"  is  the  most  important  work  that 
can  be  named  in  this  connection.  It  treats 
of  much  besides  the  teeth  of  gears,  but  its 
treatment  of  that  branch  is  particularly 
valuable.  It  is  not  easy  reading.  Wiley, 
$5.00. 

Professor  Willis'  "Principles  of  Mechan- 
ism "  is  a  celebrated  book,  now  many  years 
behind  the  age,  but  it  is,  nevertheless,  of  the 
greatest  value  and  interest  in  this  matter.  To 
Willis  we  are  indebted  for  many  of  the  most 
important  additions  to  our  knowledge  of 
theoretical  and  practical  mechanism.  Long- 
mans, $7.50.  Out  of  print. 

Professor  Rankine's  "Machinery  and  Mill- 
work  "  should  not  be  neglected  by  the 
student,  for,  although  it  is  the  dryest  of 
books,  its  value  is  as  great  as  its  reputation. 
Griffin,  $5.00. 

Professor  MacCord's  "Kinematics"  is  a 
work  that  abounds  in  novelties,  and  is  writ- 
ten in  an  attractive  style.  It  contains  many 
errors,  and  some  hobbies,  and  needs  a  thorough 
revision,  but  the  student  cannot  afford  to 
avoid  it,  or  even  to  slight  it.  Wiley,  $5.00. 

Mr.  Beale's  "Practical  Treatise  on  Gear- 
ing "  is  really  practical.  Many  of  the  so-call- 
ed "practical "  books  are  neither  practical  or 
theoretical,  but  we  have  in  this  small  book 
a  collection  of  workable  information  that 


should  be  within  the  reach  of  every  man 
who  pretends  to  be  a  machinist.  We  have 
drawn  from  it,  by  permission,  particularly 
with  regard  to  spiral  and  worm  gears. 
Mr.  Beale's  experimental  work,  in  connection 
with  the  spiral  gear,  has  been  of  great 
service.  The  Brown  &  Sharpe  Mfg.  Co., 
cloth  $1.00,  paper  75c. 

Professor  Reuleaux's  "  Konstrukteur  "  is  a 
justly  celebrated  work  in  the  German  lan- 
guage. A  translation  of  it  is  now  being 
published  in  an  American  periodical — Me- 
chanics. 

Professor  Klein,  the  translator  of  Herr- 
mann's work,  has  lately  published  the  "  Ele- 
ments of  Machine  Design,"  a  collection  of 
practical  examples,  with  illustrations.  J.  F. 
Klein,  Bethlehem,  Pa.,  $6.00. 

"Mill  Gearing,"  by  Thomas  Box,  is  a 
practical  work  by  an  engineer,  and  from  it 
we  have  drawn  much  of  our  matter  on  the 
cloudy  subject  of  the  strength  and  horse- 
power of  gearing.  Spon,  $3.00. 

"Elementary  Mechanism,"  by  Professors 
Stahl  and  Woods,  is  a  recent  work  of  general 
merit.  It  is  well  designed  as  a  text  book, 
and  treats  the  subject  in  a  simple  and  in- 
teresting manner.  Van  Nostrand,  $2.00. 

In  addition  to  the  above  works,  reference 
may  be  made  to  numerous  articles  to  be 
found  in  periodicals,  notably  in  the  "  Ameri- 
can Machinist,"  the  "Scientific  American 
Supplement,"  the  "Journal  of  the  Franklin 
Institute,"  "  Mechanics,"  and  the  "  Transac- 
tions of  the  American  Society  of  Mechanical 
Engineers." 


General   Theory . 


5. — KINEMATICS. 


This,  the  science  of  pure  mechanism,  re- 
lates exclusively  to  the  constrained  and 
geometric  motions  of  mechanism,  and  it 
has  nothing  to  do  with  questions  of  force, 
weight,  velocity,  temperature,  elasticity, 
etc.  The  path  of  a  cannon  ball  is  not  with- 
in the  field  of  kinematics,  because  it  depends 
upon  time  and  force.  A  belt  and  pulley  are 
kinematic  agents,  because  the  contact  be- 
tween them  can  be  assumed  to  be  definite, 


and  the  action  is  therefore  geometric,  but 
the  slipping  and  stretching  of  the  belt  is  not 
kinematic.  The  action  of  gear  teeth  upon 
each  other  is  purely  kinematic,  but  we  can- 
not consider  whether  the  material  is  wood,  or 
steel,  or  wax,  whether  the  gears  are  lifting 
one  pound  or  a  ton,  or  whether  they  are  run- 
ning at  one  revolution  per  second  or  one  per 
day. 


6. — ODONTICS. 


The  name  "odontics'  may  be  selected 
for  that  limited  but  important  branch  of 
kinematics  that  is  concerned  with  the  trans- 
mission of  continuous  motion  from  one 
body  to  another  by  means  of  projecting 
teeth. 

Even  this  restricted  corner  of  the  whole 
subject  is  too  large  for  the  present  purpose, 
for  it  covers  much  that  cannot  be  considered 
within  our  set  limits,  and  gear  wheels  must, 
therefore,  be  defined  as  devices  for  trans- 
mitting continuous  motion  from  one  fixed 
axis  to  another  by  means  of  engaging  teeth. 

Thus  confined,  gear  wheels  may  be  con- 
veniently divided  into  three  general  classes. 

Skew  bevel  gears,  transmitting  motion  be- 
tween axes  not  in  the  same  plane. 

Bevel  gears,  transmitting  motion  between 
intersecting  axes. 


Spur  gears,  transmitting  motion  between 
parallel  axes. 

The  last  two  classes  are  particular  cases  of 
the  first;  for,  if  the  shafts  may  be  askew  at 
any  distance,  that  distance  may  be  zero,  and 
if  they  intersect  at  any  point,  that  point  may 
be  at  infinity. 

It  would  be  scientifically  more  correct  to 
first  develop  the  skew  bevel  gear,  and  from 
that  proceed  to  the  bevel  and  spur  gear,  but 
practical  clearness  and  convenience  is  often 
more  to  be  admired  than  strict  accuracy, 
and,  as  the  true  path  is  difficult  to  follow,  we 
shall  enter  in  the  rear,  and  consider  the  spur 
gear  first. 

Odontics  does  not  properly  include  the 
consideration  of  questions  of  strength,  pow- 
er and  friction,  but  we  must  admit  certain 
important  items  in  that  direction. 


7. — PITCH 

The  fixed  axes  are  connected  with  each 
other  by  imaginary  surfaces  called  "axoids," 
or  pitch  surfaces,  touching  each  other  along 
a  single  straight  line.  We  must  imagine 
that  the  pitch  surfaces  roll  on  each  other 
without  slipping,  as  if  adhering  by  friction. 
The  whole  object  of  odontics  is  to  provide 
these  imaginary  surfaces  wilh  teeth,  by 


SURFACES. 

which  they  can  take  advantage  of  the 
strength  of  their  material  and  transmit 
power  that  is  as  definite  as  the  geometric 
motion. 

The  pitch  surface  of  the  skew  bevel  gear 
is  the  hyperboloid  of  revolution,  which  be- 
comes a  cone  when  the  axes  intersect,  and  a 
cylinder  when  the  axes  are  parallel. 


8. — NORMAL   SURFACES. 


An  important  adjunct  of  the  pitch  surface  I     For  the  skew  bevel  gear  there  does  not 


is  the  normal  surface,  or  surface  that  is 
everywhere  at  right  angles  to  both  pitch  sur- 
faces of  a  pair  of  axes,  and  upon  which  the 
action  of  the  teeth  on  each  other  may  best 
be  studied. 


appear  to  be  any  normal  surface.  For  the 
bevel  gear  the  normal  surface  is  a  sphere, 
and  for  the  spur  gear  the  sphere  becomes  a 
plane. 


General   Theory. 


9. — UNCERTAINTIES. 


The  theory  of  tooth  action  is  not  yet  full 
and  definite  in  all  its  parts,  for  there  are 
some  disputed  points,  and  some  confusion 
and  clashing  of  rules  and  systems.  This  is 


particularly  the  case  with  the  theory  of  spiral 
and  skew  bevel  teeth,  for  much  of  the  work 
that  has  been  done  is  clearly  wrong,  and 
there  is  little  that  has  been  definitely  decided. 


10.— PITCH  CYLINDERS. 


Fig.  1 


'I'itch  cylinder* 

Two  cylinders,  A  and  B,  Fig.  1,  that  will 
roll  on  each  other,  will  transmit  rotary  mo- 
tion from  one  of  the  fixed  parallel  axes  c  and 
G  to  the  other,  if  their  surfaces  are  provided 
with  engaging  projections. 

When  these  projections  are  so  small  that 
they  are  imperceptible,  the  motion  is  said  to 
be  transmitted  by  friction,  and  it  is  prac- 
tically uniform.  But  when  they  are  of 
large  size,  and  readily  observed,  the  motion, 


although  it  is  unchanged  in  nature,  is  said  to 
be  transmitted  by  direct  pressure,  and  it  is 
irregular  unless  the  acting  surfaces  of  the 
projections  are  carefully  shaped  to  produce 
an  even  motion. 

The  whole  object  of  odontics  is  to  so  shape 
these  large  projections  or  teeth  that  they 
shall  transmit  the  same  uniform  motion  be- 
tween the  rotating  cylinders,  as  would  be 
apparently  transmitted  by  friction. 

These  cylinders  are  imaginary  in  actual 
practice,  although  they  are  one  of  the 
principal  elements  of  the  theory,  and  they 
are  called  the  axoids,  or  pitch  cylinders  of 
the  gears. 

The  normal  surface  (8)  of  the  spur  gear  is 
a  plane,  and,  as  all  sections  by  normal  sur- 
faces are  alike,  we  can  study  the  action  on 
a  plane  figure  easier  than  in  the  solid  body 
of  the  gear. 


Fig.  2. 


Tooth  action 


11.— THE  LAW  OP  TOOTH   CONTACT. 

The  common  normal  to  the  tooth  curves  must 
pass  through  the  pitch  point. 

That  is,  in  Fig.  2,  if  the  tooth  curves  OD 
and  o  d  are  to  transmit  the  same  motion 
between  the  pitch  lines  pi  and  PL  as 
would  be  transmitted  by  frictional  contact 
at  the  pitch  point  0,  they  must  be  so  shaped 
that  their  common  normal  Op  at  their  com- 
mon point  p  shall  pass  through  that  pitch 
point. 

Conversely,  if  the  tooth  curves  are  so 
shaped  that  their  common  normal  always 
passes  through  the  pitch  point,  they  will 


"With  the  above  conditions  given  we  can 
deduce  the  following  law: 


transmit  the  required  uniform  motion. 


12. — THE  ODONTOID. 


This  universal  law  enables  us  to  define  the 
"odontoid,"  or  pure  tooth  curve,  for  the 
contact  of  the  pitch  lines  at  the  pitch  point 
is  continuous  and  progressive,  and,  if  the 
tooth  curves  are  to  transmit  the  same  motion, 
their  normals  must  be  arranged  in  a  contin- 


uous and  progressive  manner.  The  normals 
nl,  as  in  Fig.  3,  must  be  arranged  without 
a  break  or  a  crossing,  not  only  springing 
from  the  odontoid  at  consecutive  points,  but 
intersecting  the  pitch  line  at  consecutive 
points.  This  arrangement  may  be  called 


General   Theory. 


Fig.  3. 


Odontoid* 


"consecutive,"  and  the  definition  is  not  a 
law  by  itself,  but  an  expression  of  the  given 
universal  law. 

It  is  seen  that  the  odontoid  is  inseparably 
connected  with  its  pitch  line,  and  that  the 
same  curve  may  be  an  odontoid  with  re- 
spect to  one  pitch  line,  and  not  with  respect 
to  some  other.  The  curve  Fig.  4  is  an 
odontoid  with  respect  to  the  pitch  Mne  pi, 


but  not  with  respect  to  the  pitch  line  pF  be- 
yond the  point  p  at  which  the  normal  is  tan- 
gent to  that  pitch  line. 

The  odontoid,  so  far  as  defined,  is  not  a 
definite  thing,  and,  for  practical  purposes,  it 
must  be  given  some  particular  shape.  It 
may  be  involute  or  cycloidal,  or  of  other 
form,  but  must  always  have  normals  ar- 
ranged in  consecutive  order. 


13. — THE  LINE  OF  ACTION. 

As  the  tooth  curves  od  and  OD,  Fig.  5, 
work  together,  the  point  of  contact  will 
travel  along  a  line  Op  W  called  the  "line  of 
action." 

There  is  a  definite  relation  between  the 
odontoid  and  the  line  of  action,  so  that,  if 
either  one  is  given,  the  other  is  fixed.  If 
the  odontoid  OD  is  given,  with  its  pitch 
line  PL,  the  line  of  action  is  determined 
without  reference  to  the  pitch  .line  pi  or  its 
odontoid;  and,  conversely,  if  the  pitch  line 
and  line  of  action  are  given,  the  odontoid  to 
correspond  is  determined. 


Fig. 


Line  of 
action. 


14. — INTERCHANGEABLE  ODONTOIDS. 


This  feature  leads  at  once  to  the  broad 
and  useful  fact  that  all  odontoids,  on  pitch 
lines  of  all  sizes,  that  are  formed  from  the 
same  line  of  action,  will  work  together  inter- 
changeably, any  one  working  with  any  other. 


Therefore,  to  produce  an  interchangeable 
set  of  odontoids  we  can  choose  any  one  line 
of  action,  and  form  any  desired  number  of 
them  from  it. 


15. — INTERNAL  CONTACT. 


The  pitch  lines  of  Fig.  5  curve  in  opposite 
directions,  and  the  contact  is  said  to  be  "ex- 
ternal." But  the  principles  involved  are  in- 
dependent of  the  direction  of  the  pitch  lines, 
and  they  may  curve  in  the  same  direction,  as 
in  Fig.  6,  in  "  internal"  contact. 

Tooth  contact  is  between  lines  only,  there 
being  no  theoretical  need  of  a  solid  material 
on  either  side  of  the  line,  so  that  either  side 


Fig.  0. 


Interna.1  action 


Cusps  and   Terminah 


of  the  tooth  may  be  chosen  as  the  practical 
working  side. 

Therefore  the  internal  gear  is  precisely  like 
the  external  gear  of  the  same  pitch  diameter,  | 
working  on  the  same  lines  of  action,  so  far  j 
as  the  odontoids  are  concerned,  as  illustrated  • 
by  Fig.  7. 


Internal  and 
external  teeth 


Fig.  8 


16. — THE   CUSP  AND  INTERFERENCE. 

When,  as  in  Fig.  8,  the  pitch  circle  p  Ms 
so  small  with  respect  to  the  line  of  action 
0  C'  C"  W,  that  two  tangent  circles  G'  c'  and 
C"  c"  can  be  drawn  to  the  line  of  action  from 
the  center  G  of  the  pitch  line,  we  shall  have 
a  troublesome  convolution  in  m  the  resulting 
flank  curve  o  d.  This  convolution  will  be 
formed  of  two  cusps,  a  first  cusp  c'  on  the 
inner  tangent  arc,  the  "base  circle"  C'  c', 
and  a  second  cusp  c"  on  the  outer  tangent 
arc  G"  c". 

This  happens  with  any  form  of  odontoid, 
although  sometimes  in  disguised  form,  and 
creates  a  practical  difficulty  that  can  be 
avoided  only  by  stopping  the  tooth  curve  at 
the  first  cusp  c'. 


Furthermore,  any  odontoid  OD  that  is  to 
work  with  the  odontoid  o  d,  must  be  cut  off  j 
at  the    point  k  on    the   "limit  line"  C'  k 
through  the  point  C'  from  the  center  c. 

If  the  odontoids,  when  the  pitch  line  is  so 
small  that  the  cusps  occur,  are  not  cut  off  as 


required,  the  action  will  still  be  mathemati- 
cally perfect,  but,  as  the  contact  changes  at  a 
cusp,  from  one  side  of  the  curve  to  the  other, 
the  action  is  no  longer  practicable  with  solid 
teeth.  The  curves  will  cross  each  other, 
and  there  will  be  an  interference. 


17. — THE   SMALLEST  PITCH  CIRCLE. 


To  determine  the  smallest  pitch  circle  that 
can  be  used,  and  avoid  the  cusps  altogether, 
find  by  trial  the  point  C,  Fig.  9,  from  which 
but  one  tangent  arc  C1  c'  can  be  drawn  to  the 
line  of  action  0  C'  W.  This  point  will  be 
the  center  of  the  smallest  pitch  circle,  and 
all  points  outside  of  it  will  avoid  interference, 
while  all  inside  of  it  will  be  subject  to  it. 


Fig 


18. — THE  TERMINAL    POESTT, 

When  a  tangent  arc  can  be  drawn,  from 
the  pitch  point  0  as  a  center,  to  the  line  of 
action  at  any  point  T,  except  the  vertex  W, 
Fig.  10,  there  will  be  a  corresponding  cross- 
ing of  the  normals  to  the  odontoid  commenc- 
ing at  the  point  t,  and  a  termination  of  the 
action  when  the  point  t  reaches  the  point  T. 

As  the  action  approaches  the  terminal 
point  T  there  will  be  two  points  of  action, 


Fig.  10. 


o 

Terminal  point 


Secondary  Action. 


since  the  odontoid  crosses  the  line  of  action 
at  two  points--one  point  of  direct  and  ordi- 
nary action  at  8,  and  another  point  of  retro- 
grade and  unusual  action  at  F.  These  two 
points  of  action  will  come  together  at  T,  the 
odontoid  will  leave  the  line  of  action,  and  all 


tooth  action  will  then  cease.  The  retro- 
grade action  is  theoretically  and  actually 
correct,  but  it  is  so  oblique  that  it  is  of 
no  practical  value,  and  therefore  the  odon- 
toid may  as  well  be  cut  off  at  its  terminal 
point  t. 


19. — SPEED  OF   THE  POINT  OP  ACTION. 


Lay  off  0  8,  Fig.  5,  to  represent  the  speed 
of  the  pitch  lines,  and  draw  8  A  at  right 
angles  with  the  common  normal  0  p.  Draw 
*p  C  tangent  to  the  line  of  action  at  the  point 
of  action  p. 

Lay  off  p  B  equal  to  0  A,  and  draw  B  C 
at  right  angles  to  0  B.  Then  p  G  will  be 
the  speed  of  the  point  of  action  along  the 
line  of  action. 


When  the  line  of  action  is  a  circle  the 
angle  8  0  A  is  always  equal  to  the  angle 
B  p  C,  and  therefore  the  speed  of  the  point 
of  action  is  uniform,  and  equal  to  that  of  the 
pitch  lines. 

If  the  line  of  action  is  a  straight  line  the 
angle  B  p  G  will  be  constant — always  zero — 
and  therefore  the  speed  of  the  point  of  action 
will  be  uniform  and  always  equal  to  0  A. 


20. — NATURE  OF  THE  TOOTH  ACTION. 


The  nature  of  the  action  may  be  deter- 
mined by  a  study  of  the  normal  intersections; 
the  intersections  of  the  normals  with  the 
odontoid  being  at  uniform  distances  apart, 
their  intersections  with  the  pitch  lines  will 
indicate  the  action  of  the  teeth.  If  the  nor- 


mal intersections,  as  in  Fig.  3,  are  quite  regu- 
lar, the  action  of  the  teeth  will  be  smooth 
and  regular,  while  if  they  are  crowded  with- 
in a  narrow  space  the  action  of  the  tooth  will 
be  crowded  and  jerky. 


21. — THE  SECONDARY  LINE  OF  ACTION. 

From  the  universal  law  of  tooth  contact 
stated  in  (11)  we  can  reason  that  any 
point  on  the  tooth  curve  is  in  position  for 
contact  whenever  its  normal  passes  through 
the  pitch  point  0,  and  therefore  that  the 
point  will  then  be  upon  a  line  of  action. 

In  Fig.  11  the  normal  to  the  point  p  must 
cross  the  pitch  line  twice— at  a  primary  in- 
tersection a,  and  at  a  secondary  intersection 
b,  and  therefore  there  will  be  a  point  of 
action  on  a  primary  line  of  action  0  M'  at  q, 
when  the  curve  has  moved  so  that  the  pri- 
mary point  of  intersection  a  is  at  the  pitch 
point  0,  and  a  point  of  action  w  on  a  second- 
ary line  of  action,  when  the  secondary  point 
of  intersection  b  has  reached  the  pitch  point. 

Therefore  there  will  generally  be  not  only 
the  primary  line  of  action  0  q  M  or  O  q'  M', 
but  also  a  secondary  line  0  w  T  or  0  w'  Y1. 

The  secondary  line  of  action  must  have  the 
same  property  as  the  first,  as  a  locus  of  con- 
tact, and  therefore  if  we  can  so  arrange  two 
pitch  lines  with  their  odontoids  that  their 
secondary  lines  of  action  coincide,  there  will 
be  secondary  contact  between  the  odontoids. 


Fig 


When  it  so  happens  that  both  primary  and 
secondary  lines  coincide,  we  shall  have 
double  contact.  Two  points  of  contact  will 
exist  at  the  same  time,  one  on  the  primary 
and  the  other  on  the  secondary  line  of  action. 

The  secondary  lines  of  action  cannot  be 
made  to  coincide  when  the  contact  is  exter- 
nal, but  when  it  is  internal  they  sometimes 
can  be,  so  that  the  matter  has  an  application 
to  internal  gears. 


Interchangeable    Tooth. 


It  is  to  be  noticed  that  the  primary  line  is 
independent  of  the  pitch  line,  while  the  sec- 
ondary is  dependent  upon  it. 


Secondary  contact  is  an  interesting  feature 
of  tooth  action,  but  it  is  of  small  importance, 
and  has  been  studied  but  little. 


22. — THE  INTERCHANGEABLE   TOOTH. 


The  simple  odontoid  so  far  studied  is  the 
perfect  solution  of  the  problem  from  a 
mathematical  point  of  view,  for  it  will  trans- 
mit the  required  uniform  motion  as  long  as 
it  remains  in  working  contact.  But  from  a 
mechanical  point  of  view  it  is  still  incom- 
plete, as  it  works  in  but  one  direction, 
through  but  a  limited  distance,  and,  although 
the  odontoids  are  interchangeable,  the  gears 
are  not. 

In  order  that  the  gears  shall  be  fully  in- 
terchangeable, it  is  necessary  that  the  teeth 
shall  have  both  faces  and  flanks,  and  that  the 
line  of  action  for  the  face  shall  be  equal  to 
that  for  the  flank;  that  is,  the  tooth  must 
have  an  odontoid -on  each  side  of  the  pitch 
line,  the  face  o  d.  Fig.  12,  outside,  and  the 
flank  o  d'  inside  of  it,  and  the  line  of  action 
I  a  for  the  faces  must  be  like  the  line  of 
action  I  a'  for  the  flanks.  If  so  made,  any 
gear  will  work  with  any  other,  without  re- 
gard to  the  diameters  of  the  pitch  lines. 

But  such  a  gear  will  run  in  but  one  direc- 
tion, and  to  make  it  double-acting  it  must 
have  odontoids  facing  both  ways,  as  in  Fig. 
IS.  Gears  so  made  will  be  both  double-act- 
ing and  interchangeable,  and  it  is  not  neces- 
sary that  both  sides  of  the  tooth  shall  be 
alike. 

Again,  the  unsymmetrical  gear  of  Fig.  13 
fails  when  it  is  turned  over,  upside  down, 
for  then  the  unlike  odontoids  come  together, 
and,  to  avoid  this  last  difficulty,  all  four  of 
the  lines  of  action  must  be  alike,  producing 
the  complete  and  practically  perfect  tooth  of 
Fig.  14. 

We  can  therefore  define  the  completely  in- 
terchangeable tooth,  as  the  tooth  that  is 
formed  from  four  like  lines  of  action. 


Unsynt-inetrical  teeth 

Fig.  14. 


Complete  teeth 


&. — INTERCHANGEABLE  RACK   TOOTH. 


When  the  pitch  line  is  a  circle  the  flanks  of 
the  tooth  are  not  like  the  faces,  but  when  it 
is  a  straight  line  there  is  no  distinction  be- 


tween face  and  flank.  We  then  have  the  im- 
portant practical  fact  that  the  four  odontoid^ 
of  the  interchangeable  rack  tooth  are  alike. 


Construction  by  Points. 


24. — CONSTRUCTION   BY  POINTS. 

When  we  have  an  odontoid  and  its  pitch 


line  given,  it  is  a  very  simple  matter  to  con- 
struct either  the  line  of  action  or  the  conju- 
gate odontoid  for  any  other  pitch  line. 

We  know,  for  example,  the  odontoid  «  p, 
Fig.  15,  on  the  pitch  line  p  I,  and  it  is  re- 
quired to  construct  an  odontoid  on  the  pitch 
line  P  L  that  is  conjugate  to  it. 

As  the  odontoid  is  given  we  know  or  can 
construct  its  normals.  Construct  the  normal 
p  a  from  any  chosen  point  p,  draw  the  radial 
line  da  C,  lay  off  A  0  equal  to  a  0,  draw  the 
radial  line  A  C,  lay  off  the  angle  NAD 
equal  to  the  angle  n  a  d,  lay  off  P  A  equal 
to  p  a,  and  P  will  be  a  point  in  the  required 
conjugate  odontoid  8  P.  P  A  will  be  a 
normal  to  the  curve.  Construct  a  number  of 
points  by  this  process,  and  draw  the  required 
curve  through  them.  The  tangents  *  t  and 
8  T  make  equal  angles  with  the  pitch  lines, 
so  that  the  required  curve  can  often  be  fully 
determined  by  drawing  its  tangent  and  one 
or  two  points. 

To  construct  the  line  of  action,  make  the 
angle  m  0  e  equal  to  the  angle  n  a  d, 
and  lay  off  0  q  equal  to  p  a.  The  point 
q  is  on  a  circle  from  either  p  or  P  drawn 
from  the  centers  (7,  and  is  the  point  at 
which  p  and  P  will  coincide  when  the  two 
curves  are  in  working  contact,  the  normals 
p  a  and  P  A  then  coinciding  with  the 
radiant  0  q. 


Fig.  15. 


Construrtio 
by  points 


When  the  line  of  action  alone  is  given,  the 
odontoids  for  given  pitch  lines  are  fully  de- 
termined, but  there  seems  to  be  no  simple 
graphical  method  for  constructing  them  ex- 
cept for  special  cases.  They  can  be  obtained 
by  the  use  of  the  calculus  (33),  or  drawn  by 
the  integrating  instrument  of  (34). 

The  two  tooth  curves  thus  constructed  are 
paired,  and  are  said  to  be  "  con  jugate'*  to 
each  other. 


25.— THE  ABC  OP  ACTION. 

The  action  between  two  teeth  commences 
and  ends  at  the  intersections  m  and  N  of  the 
line  of  action  with  the  addendum  lines  of  the 
two  gears,  a  I  and  A  L,  Fig.  16.  The  arc 
of  action  is  the  distance  a  b  on  the  pitch  line 
that  is  passed  over  by  the  tooth  while  it  is  in 
action. 

The  arc  a  0  passed  over  while  the  point  of 
contact  is  approaching  the  pitch  point,  is 
called  the  arc  of  approach,  and  0  b,  that 
passed  over  while  the  action  is  receding  from 
that  point,  is  the  arc  of  recess. 

With  a  given  line  of  action  the  arcs  of  ap- 
proach and  recess  can  be  controlled  by  the 
addenda.  If  it  is  desirable  to  have  a  great 
recess  and  a  small  approach,  the  addendum 


of  the  gear  that  acts  as  a  driver  is  to  be  in- 
creased. When  there  is  a  limit  line  (16),  it 
limits  the  addendum  and  the  arc  of  action. 


10 


Molding  Processes.        ^ 


26.— OBLIQUITY 

When  a  pair  of  teeth  bear  upon  each 
other,  the  direction  of  the  force  exerted  be- 
tween them  is  that  of  the  common  normal 
Op,  Fig.  17,  and  passes  through  the  pitch 
point  0.  Except  when  the  point  of  contact 
Is  at  the  pilch  point  the  direction  of  the 
pressure  will  deviate  from  the  normal  to  the 
line  of  centers  by  the  angle  of  obliquity 
Z  Op,  and  with  many  forms  of  teeth  the 
angle  is  never  zero. 

The  force  exerted  between  two  teeth  at 
their  point  of  contact  is  found  by  laying  off 
the  tangential  force  0  //with  which  the  driv- 
ing gear  D  is  turning,  and  drawing  the  line 
H  V  parallel  to  the  line  of  centers,  to  find 
the  force  0  V  —  P  K.  It  is  proportional  to 
ihe  secant  of  the  angle  of  obliquity,  and  in- 
creases rapidly  with  that  angle. 

The  chief  influence  of  the  obliquity  is 
upon  the  friction  between  the  teeth,  and  con- 
sequent inefficiency  of  the  gear,  and  upon 
the  destruction  by  wearing.  It  is  par- 
ticularly important  upon  the  approaching 
action,  and  a  gear  that  is  otherwise  perfect 
may  be  inoperative  on  account  of  excessive 
obliquity. 

Although  the  direct  pressure  of  the  teeth 
upon  each  other  at  their  point  of  contact 


OF    THE   ACTION 
K 


.17. 

will  vary  with  the  obliquity,  the  tangential 
force  exerted  to  turn  the  gear  is  always 
uniform.  Leaving  friction  out  of  the  calcu- 
lation, the  two  gears  of  a  pair  always  turn 
with  the  same  force  at  their  pitch  lines. 

The  obliquity  of  the  action  has  an  effect 
upon  the  direction  and  amount  of  the 
pressure  of  the  gear  upon  its  shaft  bearing, 
but  the  usual  variation  is  of  little  conse- 
quence. 

It  is  desirable  that  the  pressure  between 
the  teeth  should  be  as  uniform  as  possible, 
not  only  in  amount,  but  in  direction,  and 
excessive  obliquity  is  to  be  carefully  avoided. 


27.— CONSTRUCTION  BY  MOLDING. 


The  mode  of  action  of  the  conjugate  teeth 
upon  each  other,  suggests  a  process  by  which 
a  given  tooth  can  be  made  to  form  its  conju- 
gate by  the  process  of  molding. 

The  given  tooth,  all  of  its  normal  sections 
being  of  some  odontoidal  form,  is  made 
of  some  hard  substance,  while  the  blank  in 
which  the  conjugate  teeth  are  to  be  formed 
is  made  of  some  plastic  material.  The  shafts 
of  the  two  wheels  are  given,  by  any  means, 
the  same  motions  as  if  their  pitch  surfaces 
were  rolled  together.  The  hard  tooth  will 
then  mold  the  soft  tooth  into  the  true  conju- 
gate shape. 


It  matters  not  what  shape  is  given  the 
molding  tooth,  if  its  sections  are  all  odon- 
toidal, and  a  twisted  or  irregular  shape  will 
be  as  serviceable  as  the  common  straight  tooth. 

This  process  is  continually  in  operation  be- 
tween a  pair  of  newly  cut  teeth,  or  between 
rough  cast  teeth,  until  the  badly  matched 
surfaces  have  been  worn  to  a  better  fit,  but 
it  is  too  slow  for  ordinary  purposes,  and  is 
of  little  practical  value. 

Gears  can  be  formed  by  this  process,  by 
rolling  a  steel  forming  gear  against  a  white 
hot  blank,  but  the  process  can  hardly  be 
called  practical. 


28.— MOLDING   PLANING    PROCESS. 


Although  the  described  molding  process 
is  of  limited  practical  value,  having  but  one 
direct  application,  it  leads  to  a  process  of 
great  value  when  the  tooth  is  straight  or  of 


such  a  shape  that  it  can  be  followed  by  a 

planing  tool,  its  normal  sections  being  alike. 

The  originating  tooth  is  fixed  in  the  shape 

of  a  steel  cutting  tool  C,  Fig.  18,  which  is 


Planing  Processes. 


\  \ 


rapidly  reciprocated  in  guides  G,  in 
the  direction  of  the  length  of  the 
tooth,  as  the  two  pitch  wheels  A  and 
B  are  rolled  together.  Although  the 
tool  has  but  a  single  cutting  edge,  its 
motion  makes  it  ihe  equivalent,  of  the 
molding  tooth,  and  it  will  plane  out 
the  conjugate  tooth  D  by  a  process 
that  is  the  equivalent  of  the  more 
general  molding  process. 

A  simple  graphical  method  is 
founded  upon  this  molding  process, 
the  shaping  tool  taking  the  form  of  a 
thin  template  (7,  Fig.  19,  that  is  re- 
peatedly scribed  about  as  the  pitch 
wheels  are  rolled  together,  the  marks 
combining  to  form  the  conjugate 
tooth  curves  D. 

This  mechanical  process  has  the 
decided  advantage  over  the  procets 
of  construction  by  points  (24),  that 
the  tooth  is  formed  with  a  correct 
fillet  (44),  and  is  much  stronger. 
The  dotted  lines  show  the  tooth  that 
would  be  constructed  by  points. 

The  only  practicable  method  for 
forming  the  line  of  action  when  this 
method  is  used  is  by  observing  and 
marking  a  number  of  points  of  con- 
tact between  the  teeth.  This  method 
is  applicable  to  all  possible  forms  of 
spur  teeth,  either  straight,  twisted  or 
spiral.  It  can  be  practically  applied 
only  to  the  octoid  form  of  bevel  tooth. 

On  account  of  the  fillet  (44)  that 
is  formed  by  this  process,  the  tooth 
space  cannot  be  used  with  a  mating 
gear  having  more  teeth  than  that  of 
the  forming  gear,  although  it  belongs 
to  the  same  interchangeable  set.  The 
tooth  space  of  the  figure  will  not  run 
with  a  tooth  on  a  pitch  line  larger 
than  the  pitch  line  A. 

Therefore  the  rack  tooth  must  be 
useu  as  the  forming  tooih,  to  allow  of 
the  use  of  all  gears  of  the  set  up  to 
the  rack.  Gears  of  the  set  thus  formed 
will  not  work  with  internal  gears. 


Hlolding  planing 
method 


fttcipro  eating 
Too/  >- 


JFig.  18. 


phical  molding 
method 


29 — LINEAR     PLANING     PROCESS. 


A  second  planing  process,  quite 
distinct  from  the  molding  process  of 
(27),  is  founded  upon  the  fact  that 
the  tooth  curves  are  in  contact  at  a 
single  point  which  has  a  progressive 
motion  along  the  line  of  action. 

Therefore  if  a  single  cutting  point 
p,  Fig.  20,  is  caused  to  travel  along 
the  line  of  action  with  the  proper 
speed  relatively,  to  the  speed  of  the 
piich  line,  it  will  trim  the  tooth  out- 
line to  the  proper  odontoidal  shape. 

The  figure  shows  the  application  to 


Fig.  20. 

fn^olu-^-e  To 


Linear  plan  ing 
met 


' 

OF  THZ 


12 


Particular  Forms. 


the  involute  tooth,  the  path  of  the  cutting 
point  being  the  straight  line  I  a,  and  its  speed 
being  the  speed  of  the  base  line  b  I. 

When  the  cutting  point  follows  the  circu- 
lar line  of  action  with  a  speed  equal  to  that 
of  the  pitch  line,  it  will  plane  out  the 
cycloid al  tooth  curve. 


This  process  is  applicable  to  all  possible 
forms  of  gear  teeth,  either  spur  or  bevel,  in 
either  external  or  internal  contact. 

When  the  curvature  of  the  odontoid  will 
permit,  the  milling  cutter  may  take  the  place 
of  the  planing  tool,  and  is  the  equivalent  of  it. 


). — THE    BACK    ORIGINATOR. 


The  molding  planing  process  of  (28)  sup- 
plies a  means  for  easily  and  accurately  pro 
ducing  an  interchangeable  set  of  gears  or 
cutters  for  gears,  and  it  is  best  applied  by 
means  of  the  rack  tooth  as  the  originator. 
All  four  curves  of  the  rack  tooth  being  alike, 
the  tooth  is  easily  formed,  particularly  for 
the  involute  or  the  segmental  systems,  and  it 
is  a  matter  of  less  consequence  that  the  curves 


shall  be  of  some  particular  form,  if  care  is 
taken  that  it  is  odontoidal. 

It  has  been  taught,  and  it  is  therefore  some- 
times considered,  that  any  "  four  similar  and 
equal  lines  in  alternate  reversion"  will  an- 
swer the  purpose,  but  it  is  necessary  that  the 
four  similar  curves  shall  be  odontoids.  Four 
circular  arcs,  with  centers  on  the  pitch  line,  will 
answer  the  definition,  but  are  not  odontoids. 


31.— PARTICULAR  FORMS  OF  THE  ODONTOID. 


The  odontoid,  as  so  far  examined,  is  un- 
defined except  as  to  one  feature  of  the  ar- 
rangement of  its  normals,  and  to  bring  it 
into  practical  use  it  is  necessary  to  give  it 
some  definite  shape.  This  is  most  easily  ac- 
complished by  choosing  some  simple  curve 
for  the  rack  odontoid,  and  from  that  making 
an  interchangeable  set.  A  more  correct  but 
much  more  difficult  method  would  be  to 
choose  some  definite  line  of  action,  and  from 
that  derive  the  odontoids. 

If  the  rack  odontoids  are  straight  lines, 
Fig.  21,  the  common  involute  tooth  system 
will  be  produced,  and  the  line  of  action  will 
be  a  straight  line  at  right  angles  with  the 
rack  odontoid.  For  bevel  teeth,  as  will  be 
shown,  the  straight  line  odontoid  produces 
the  octoid  tooth  system,  while  to  produce  the 
involute  system  it  is  necessary  to  define  the 
line  of  action  as  a  straight  line,  and  derive 
the  system  from  that. 

If  the  rack  odontoids  are  cycloids,  as  in 
Fig.  22,  the  resulting  tooth  system  will  be 
ihe  cycloidal,  commonly  misnamed  the 
"  epicycloidal  "  system.  The  line  of  action 
will  be  a  circle  equal  to  the  roller  of  the 
cycloid. 

If  the  rack  odontoids  are  segments  of  cir- 
cles from  centers  not  on  the  pitch  line,  but 
inside  of  it,  as  in  Fig.  23,  the  tooth  system 


Fig 


Segmental 

Fig.  23. 


Rolled   Curve    Theory. 


will  be  the  segmental,  and  its  line  of  action 
will  be  the  loop  of  the  "  Conchoid  of  Nico- 
medes." 

If  we  choose  a  parabola  for  the  rack  tooth, 
as  in  Fig.  24,  the  parabolic  system  will  be 
formed  with  its  peculiar  "hour  glass"  line 
of  action. 

Only  three  of  these  tooth  systems  are  in 
actual  use,  the  involute  and  the  cycloidal  for 
spur  gears,  and  the  octoid  for  bevel  gears 
only,  and  we  will  therefore  confine  the  ap- 
plication of  the  theory  to  them. 

Only  one  of  the  sys'ems  in  common  use 
for  spur  gears,  the  involute,  should  be  in  use 
at  all,  and  we  will  pay  principal  attention  to 
that. 


Parabolic 

Fig 


The  segmental  system  would  be  superior 
to  the  cycloidal,  and  in  many  cases  to  the  in- 
volute; but  as  there  is  already  one  system  too 
many,  we  will  not  attempt  to  add  another. 


32.— THE    ROLLED     CURVE     THEORY. 


If  any  curve  R,  Fig.  25,  is  rolled  on  any 
pitch  curve  p  I,  a  point  p  in  the  former  will 
trace  out  on  the  plane  of  the  latter  a  curve 
s  p  z,  called  a  rolled  curve. 

The  line  p  q,  from  the  tracing  point  p  to 
the  point  of  contact  q,  is  a  normal  to  the 
curve  *  p  z,  and,  as  all  the  normals  are  ar- 
ranged in  "consecutive"  oraer,  that  curve 
must  be  an  odontoid.  The  converse  of  this 
statement  is  also  true,  that  all  odontoids  are 
rolled  curves  ;  but  the  fact  is  general^  .  ery 
far  fetched  and  of  no  practical  importance. 

It  is  also  a  property  of  all  such  curves 
that  are  rolled  on  different  pitch  lines,  that 
they  are  interchangeable. 

This  accidental  and  occasionally  useful 
feature  of  the  rolled  curve  has  generally 
been  made  to  serve  as  a  basis  for  the  general 
theory  of  the  gear  tooth  curve,  and  it  is  re- 
sponsible for  the  usually  clumsy  and  limited 
treatment  of  that  theory.  The  general  law 
is  simple  enough  to  define,  but  it  is  so  diffi- 
cult to  apply,  that  but  one  tooth  curve,  the 
cycloidal,  which  happens  to  have  the  circle 
for  a  roller,  can  be  intelligently  handled 
by  it,  and  the  natural  result  is,  that  that 
curve  has  received  the  bulk  of  the  atten- 
tion. 

For  example,   the    simplest    and  best  of 


Rolled  curve 

Fig.  25. 

all  the  odontoids,  the  involute,  is  entirely 
beyond  its  reach,  because  its  roller  is  the 
logarithmic  spiral,  a  transcendental  curve 
that  can  be  reached  only  by  the  higher  mathe- 
matics. 

No  tooth  curve,  which,  like  the  involute, 
crosses  the  pitch  line  at  any  angle  but  a 
right  angle,  can  be  traced  by  a  point  in  a 
simple  curve.  The  tracing  point  must  be 
the  pole  of  a  spiral,  and  therefore  the  trac- 
ing of  such  a  curve  is  a  mechanical  impossi- 
bility. A  practicable  rolled  odontoid  must 
cross  the  pitch  line  at  right  angles. 

To  use  the  rolled  curve  theory  as  a  base  of 
operations  will  confine  the  discussion  to  the 
cycloidal  tooth,  for  the  involute  can  only  be 
reached  by  abandoning  its  true  logarithmic 
roller,  and  taking  advantage  of  one  of  its 
peculiar  properties,  and  the  segmental, 
sinusoidal,  parabolic,  and  pin  tooth,  as  well 
as  most  other  available  odontoids,  cannot  be 
discussed  at  all. 


33. — MATHEMATICAL  RELATION   OP   ODONTOID   AND   LINE   OF   ACTION. 

In  Fig.  26  the  odontoid  on  the  pitch  line    by  the  relations  P  T  =  p  t  =  y,  and  T  8  — 
p  I  is  connected  with  the  line  of  action  I  a,  j  t  0  =  x,  where  P  8  is  the  normal  to  the 


14 


Mathematical  Relation. 


odontoid  at  the  point  P,  T  8  is  a  tangent  to 
the  pitch  line  at  the  intersection  of  the  nor- 
mal, and  P  TMsa  normal  to  the  tangent. 

When  any  odontoid  is  given  by  its  equa- 
tion, that  of  the  line  of  action  can  be  found 
by  a  process  of  differentiation,  and  when 
the  line  of  action  is  given  by  its  equation, 
that  of  the  odontoid  can  be  found  by  a 
process  of  integration. 

These  processes,  for  the  general  case  where 
the  pitch  line  is  curved,  are  quite  intricate, 
Imt  when  the  pitch  line  is  a  straight  line, 
they  are  simple,  and  may  be  worked  as 
follows. 

To  get  the  equation  of -the  line  of  action 
from  that  of  the  given  rack  odontoid,  ar- 
Tange  the  equation  of  the  odontoid  in  the 
form  x  —  f(y),  and  put  its  differential  co- 
efficient - —  equal  to  — .  Thus,  the  equation 
d  y  x 

of  the  straight  rack  odontoid  of  the  involute 
system  is  y  —  x  tan.  A,  from  which 
<lx  1  y  x 

Tj  ^  't^TA  '•  £•"*»  =tST^i  1S 

the  equation  of  the  straight  line  of  action  at 
right  angles  to  the  odontoid.  Again,  the 
equation  of  the  cycloid  being  x  =  vi  r.  sin.-1 
x  =  ver.  sin.  -' 


Fig.  2V. 

and  xa  -\-ya  —  2ry  is  the  equation  of  the 
circular  line  of  action. 

To  get  the  equation  of  the  odontoid  when 
that  of  the  line  of  action  is  given,  arrange 
the  equation  of  the  line  of  action  in  the  form 

—  =f(y)f  put  it  equal  to  -  - — ,  and  inte- 
grate. T.hus,  the  equation  of  the  straight 
line  of  action  being 

x 

lan.  A  ' 
we  have 

y  1         _  dx 

x          tan.  A     ~  dy' 

and  y  —  x  tan.  A  is  the  equation  of  the 
straight  odontoid  at  right  angles  to  the  line 
of  action.  Again,  the  equation  of  the  circu- 
lar line  of  action  being  x*  -|-  y2  =  2ry,  we 
have 


and  x  = 
cycloidal  odontoid. 


34. — THE    ODONTOIDAL 

The  form  <5f  the  odontoid  to  correspond  to 
&  'given  line  of  action  and  a  given  pitch  line 
can  be  determined  only  by  the  integral  cal- 
culus (33),  it  evidently  being  impossible  to 
contrive  a  general  graphical  or  algebraic 
method. 

But  it  can  be  directly  drawn  by  an  instru- 
ment, the  principle  of  which  is  analogous  to 
that  of  the  well-known  polar  planimeter  for 
Integrating  surfaces. 

The  bar  jR,  Fig.  27,  moves  at  right  angles 
to  the  line  of  centers,  and  it  moves  the  pitch 
•wheel  A,  with  the  same  speed  at  the  pitch 
line.  The  bar  G  has  a  point  p,  that  is 
•confined  to  move  in  the  given  line  of  action 
Op  W,  and  it  is  so  guided  that  it  always 
passes  through  the  pitch  point  0. 

The  two  bars  bear  upon  each  other  by 
friction,  and  we  must  suppose  that  there 
5s  no  other  friction  to  oppose  the  motion  of 
the  bar  C. 


Odontoidal  Integrate? 


Then  the  pointy  will  trace  out  the  odontoid 
spzupon  the  pitch  wheel  A,  or  upon  any 
other  pitch  wheel  B  rolling  with  the  bar  R 
on  either  side  of  it. 


THE    SPUR.    GEAR    IN     GENERAL. 

35. — THE  CIRCULAR    PITCH. 


The  distance  a  0,  Fig.  14,  covered  by  each 
tooth  upon  the  pitch  circle,  is  commonly 
called  the  "circular  pitch,"  and  often  the 
"circumferential  pitch."  The  term  "pitch 
arc"  is  the  most  appropriate  but  is  not  in 
common  use. 

This  was  formerly  the  measurement  by 
which  the  size  of  the  tooth  was  always 
stated,  a  tooth  being  said  to  be  of  a  certain 
"  pitch,"  and  all  of  its  other  dimensions 
being  expressed  in  terms  of  that  unit,  but  it 
is  fast  being  replaced,  and  should  be  entirely 
replaced,  by  the  more  convenient  "diametral 
pitch  "  unit. 

The  circumference  of  a  circle  is  measured 
in  terms  of  its  diameter  by  means  of  an  in- 
commensurable fractional  number  3.14159, 
called  TT  (pi),  and,  therefore,  if  the  tooth  is 
measured  upon  the  arc  of  the  circle  by  means 
of  the  circular  pitch,  one  of  two  inconveni- 
ences must  be  tolerated.  Either  the  pitch 
must  be  an  inconvenient  fraction,  or  else  the 
pitch  diameter  must  be  as  inconvenient,  for 
the  gear  cannot  have  a  fractional  number  of 
teeth.  The  fractional  calculations  are  so 
clumsy  that  a  table  of  pitch  diameters  cor- 
responding to  given  numbers  of  teeth  should 
be  used,  and  errors  in  the  laying  out  of  the 
work  are  of  constant  occurrence. 

Again,  outside  of  the  liability  of  error  in 
making  calculations,  the  circular  pitch  sys- 
tem is  a  constant  source  of  error  in  the  hands 
of  lazy  or  incompetent  draftsmen  or  work- 
men, for  there  is  a  constant  temptation, 
often  yielded  to,  to  force  the  clumsy  figures 
a  little  to  produce  some  desired  result.  For 
example,  a  millwright  has  to  make  a  gear  of 
fourteen  inches  pitch  diameter  with  fourteen 
teeth.  He  finds  by  the  usual  computation 
that  the  circular  pitch  is  3.14  inches,  and,  as 
his  odontograph  has  a  table  for  three-inch 
pitch,  he  uses  that  with  the  remark  that  it  is 
"near  enough,"  laying  the  blame  on  the 
odontograph  or  on  the  iron  founder  if  the 
resulting  gear  roars.  His  next  order  is  for  a 


gear  of  one-inch  pitch  to  match  others  in 
use,  and  to  be  fourteen  and  a  half  inches 
diameter.  The  circumference  of  the  pitch 
line  is  45.53  inches,  and  he  has  his  choice  be- 
tween 45  and  46  teeth,  both  wrong.  Per- 
haps the  most  frequent  cause  of  error  is 
that  the  workman  is  apt  to  apply  a  rule 
directly  to  the  teeth  of  a  gear  he  is  about  to 
repair  or  match,  to  get  the  circular  pitch, 
and  the  result  is  more  likely  to  be  wrong  than 
right. 

The  best  plan  when  using  this  unit  is  to 
get  convenient  pitch  diameters  and  let  the 
pitch  come  as  it  will,  provided  that  gears 
that  work  together  are*of  the  same  pitch,  and 
that  is  simply  a  roundabout  way  of  using  the 
diametral  pitch  unit. 

When  tbe  circular  pitch  must  be  used  the 
following  table  will  greatly  assist  the  work 
and  save  calculation.  For  example,  the 
pitch  diameter  of  a  gear  of  three-quarter-inch 
pitch  and  37  teeth  is  three-quarters  the  tabu- 
lar number  11.78,  or  8.84  inches. 

PITCH    DIAMETERS. 
FOR  ON/5  INCH  CIRCULAR  PITCH. 

FOR  ANY  OTHER  PITCH   MULTIPLY  BY  THAT  PITCH. 


T.   P.  D. 

T.  P.  D. 

T.  P.  D. 

T.  P.  D. 

10 

8.18 

33 

10.50 

58 

17.83 

79 

25.15 

11 

3.50 

34 

10.82 

57 

18.14 

80 

25.47 

12 

3  H2 

35 

11.14 

58 

18.46 

81 

25.79 

13 

4.14 

36 

11.46 

59 

18.78 

82 

26.10 

14 

4.46 

37 

11.78 

60 

19.10 

83 

26.42 

15 

4.78 

38 

12.10 

61 

19.42 

84 

26.74 

16 

5  09 

39 

12.42 

62 

19.74 

85 

27.06 

17 

5.41 

40 

12.73 

63 

20.06 

86 

27.38 

18 

5.73 

41 

13.05 

64 

20.37 

87 

27  70 

19 

6.05 

42 

13.37 

65 

20.69 

88 

28.01 

20 

6.37 

43 

13.69 

66 

21.01 

89 

2.S.33 

21 

6.69 

44 

14.00 

67 

21.33 

90 

28.65 

22 

7.00 

45 

14.33 

68 

21.65 

91 

28  97 

m 

7.32 

46 

14  64 

69 

21.97 

92 

29.29 

24 

7.64 

47 

14.96 

70 

22.28 

93 

29.60 

25 

7.96 

48 

15.28 

71 

22  60 

94 

29.92 

26 

8.28 

49 

15.60 

72 

22.92 

95 

30.24 

27 

8.60 

50 

15.92 

73 

23.24 

96 

30.56 

28 

8.91 

51 

16.24 

74 

23.56 

97 

30.88 

29 

9.23 

52  16.55 

75 

23.88 

98 

31.20 

30 

9.55 

53  16.87 

76 

24.19 

99 

31.52 

31 

0.87 

54  17.19 

77 

24.51 

100 

31.83 

32 

1 

10.19 

55  j  17.51 

78 

24.  83 

36. — THE  DIAMETRAL  PITCH. 


This  is  not  a  measurement,  but  a  ratio 
or  proportion.  It  is  the  number  of  teeth  in 
the  gear  divided  by  the  pitch  diameter  of  the 


gear.  Thus,  a  gear  of  48  teeth  and  12  inches 
pitch  diameter  is  of  4  pitch.  The  advantages 
of  the  diametral  pitch  unit  are  so  apparent 


1C 


Pitches  and  Addendum. 


that  it  is  fast  displacing  the  circular  pitch 
unit,  and  has  almost  entirely  displaced  it  for 
cut  gearing.  It  is  so  simple  that  a  table  of 
pitch  diameters  is  entirely  useless,  although 
such  useless  tables  have  been  published. 


The  diametral  pitch  is  sometimes  defined  as 
the  number  of  teeth  in  a  gear  of  one  inch 
diameter.  It  is  a  common,  but  bad  practice, 
to  designate  diametral  pitches  by  numbers, 
as  No.  4,  No.  16,  etc. 


Diametral 

Circular 

Circular 

Diametral 

37.  —  RELATION   OF  PITCH   UNITS. 

Pitch. 

Pitch. 

Pitch. 

Pitch. 

The  product  of  the  circular  pitch  by  the 

2 

1.571  inch 

2 

1-571 

diametral    pitch    is   the    constant    number 

1.396. 
1.257 

1  IS76 
1  795 

3.1416,  so  that  if  one  is  given  the  other  is 

2% 

1J42 

i4 

easily  calculated. 

3 

1.047 
.898 

? 

2  094 
2  185 

The  following  tables  of  equivalent  pitches 

4 

5 

.785 

K 

2  2f<5 
2.394 

will  be  convenient  in  this  connection. 

6 

'.524 

y. 

2^513 

7 

.449 

3ff 

2.646 

8 

.393 

IT& 

2.793 

9 

.349 

IT'S 

2.957 

10 

.314 

1 

3.142 

. 

11 

.288 

15 

3  351 

12 

.262 

4? 

3  590 

38.—  ACTUAL   SIZES. 

14 

!224 

n 

3  '.867 

16 

.196 

A^ 

4.189 

Figs.  28  and  29  show  the  actual  sizes  of 

18 

.175 

\b 

4.570 

standard  teeth  of  the  usual  diametral  pitches, 

20 
22 

.157 
.143 

A 

5.027 
5.  £88 

and  give  a  better  idea  of  the  actual  teeth  than 

24 
26 

.131 
.121 

8 

6.283 

7  181 

can  be  given  by  any   possible  description. 

28 

.112 

5 

8  378 

They  are  printed  from  cut  teeth,  and  may  be 

30 
32 

.105 

.098 

il 

10  Of>3 
12.566 

depended  upon  as  accurate. 

36 
40 

.087 
.079 

ft 

16.755 
25.133 

48 

.065 

T15 

50.266 

39.— ADDENDUM   AND  DEDENDUM. 


The  tooth  is  limited  in  length  by  the  circle  j 
a  I,  Fig.  30,  called  the  addendum  line,  and 
drawn  outside  the  pitch  line  at  a  given 
distance,  called  the  addendum.  Its  depth  is 
also  limited  by  aline  r  I,  called  the  dedendum 
or  root  line,  drawn  at  a  given  distance  inside 
of  the  pitch  line. 

The  addendum  and  the  dedendum  are 
both  arbitrary  distances,  but,  for  convenience 
in  computation,  they  are  nxed  at  simple 
fractions  of  the  unit  of  pitch  that  is  in  use. 
When  the  circular  pitch  is  used  the  ad> 
dendum  is  one-third  of  the  circular  pitch. 

When  the  diametral  pitch  unit  is  used 
the  addendum  is  one  divided  by  the  pitch. 

It  is  customary  to  make  the  addendum  and 
the  dedeodum  the  same,  except  in  certain 
cases  where  some  special  requirement  is  to 
be  satisfied. 


Fig.  30. 


#1 


Addenda 
Clearance 
Backlash 


Actual  Sizes. 


17 


18 


Actual  Sizes. 


2  Pitch. 


2  I  Pitch. 


3  Pitch. 


Fig.  29. 


Items  of  Construction. 


19 


40.— THE   CLEARANCE. 


To  allow  for  the  inevitable  inaccuracies  of 
workmanship,  especially  on  cast  gearing,  it 
is  customary  to  carry  the  tooth  space  slightly 
below  the  root  line  to  the  clearance  line  c  I, 
Fig.  30. 


The  clearance,  or  distance  of  the  clearance 
line  inside  of  the  root  line,  is  arbitrary,  but  it 
is  convenient  and  customary  to  make  it  one> 
eighth  of  the  addendum. 


41. — THE   BACK-LASH. 


When  rough  wooden  cogs  or  cast  teeth  are 
used,  the  irregularities  of  the  surface,  and 
inaccuracies  of  the  shape  and  spacing  of 
the  teeth,  require  that  they  should  not  pre- 
tend to  fit  closely,  but  that  they  should  clear 
each  other  by  an  amount  b,  Fig.  30,  called 
the  back-lash. 

The  amount  of  the  back-lash  is  arbitrary, 


but  it  is  a  good  plan  to  make  it  about  equal 
to  the  clearance,  one-eighth  of  the  addendum. 
Skillfully  made  teeth  will  require  less, 
back-lash  than  roughly  shaped  teeth,  and". 
properly  cut  teeth  should  require  no  back- 
lash at  all.  Involute  teeth  require  less  back- 
lash than  cycloidal  teeth. 


42. — THE   STANDARD   TOOTH. 


The  tooth  must  be  composed  of  odontoids, 
preferably  of  odontoids  of  which  the  proper- 
ties are  well  known,  and  an  advantage  is 
gained  if  it  is  still  further  confined  to  a  par- 
ticular value  of  that  odontoid.  If  the  teeth 
are  to  be  drawn  by  an  odontograph  some 
standard  must  be  fixed  upon,  since  the 


method  will  cover  but  one  proportion  of  tooth.. 
For  example,  the  standard  involute  tooth 
is  that  having  its  line  of  action  inclined  at  an 
angle  of  obliquity  of  fifteen  degrees.  For 
the  cycloidal  system  the  standard  agreed  upon 
is  the  tooth  having  radial  flanks  on  a  gear  of 
twelve  teeth. 


43 .  — ODONTOGR  APHS . 


The  construction  of  the  tooth  is  generally 
not  simply  accomplished  by  graphical  means, 
as  it  is  generally  required  to  find  points  in 
the  curve  and  then  find  centers  for  circular 
arcs  that  will  approximate  to  the  curve  thus 
laid  out. 

It  is  sometimes  attempted  to  construct  the 
curve  by  some  handy  method  or  empirical 


rule,  but  such  methods  are  generally  worth- 
less. 

An  odontograph  is  a  method  or  an  instru- 
ment for  simplifying  the  construction  of  the 
curve,  generally  by  finding  centers  for  ap- 
proximating circular  arcs  without  first  find- 
ing points  on  the  curve,  and  those  in  use  will 
be  described. 


44. — THE 

When  the  teeth  are  laid  out  by  theory 
there  will  be  a  portion  of  the  tooth  space  at 
the  bottom  that  is  never  occupied  by  the 
mating  tooth.  Fig.  31  shows  a  ten-toothed 
pinion  tooth  and  space  with  a  rack  tooth  in 
three  of  its  positions  in  it,  showing  the  un- 
used portion  by  the  heavy  dotted  line. 

If  this  unused  space  is  filled  in  by  a 
"  fillet "/  the  tooth  will  be  strengthened  just 
where  it  needs  it  the  most,  at  the  root. 

The  fillet  is  dependent  on  the  mating  tooth, 
and  is  therefore  not  a  fixed  feature  of  the 
tooth.  If  a  gear  is  to  work  in  an  inter- 
changeable set,  it  may  at  some  time  work 
with  a  rack,  and  therefore  its  fillet  should  be 
fitted  to  the  rack ;  but  if  it  is  to  work  only 


FILLET. 

with  some  one  gear  it  may  be  fitted  to  that. 
The  light  dotted  line  shows  the  fillet  that 
would  be  adapted  to  a  ten-toothed  mate. 
The  fillet  to  match  an  internal  gear  tooth 
would  be  even  smaller  than  that  made  by  the 
rack. 


31. 


The  fillet 


20 


Equidistant   Series. 


When  the  tooth  is  formed  by  the  molding 
process  of  (27),  or  by  the  equivalent  planing 
process  of  (28),  the  fillet  will  be  correctly 
formed  by  the  shaping  tool,  but  not  so  when 
the  linear  process  of  (29)  is  used.  When  the 
tooth  is  drawn  by  theory  or  by  an  odonto- 
graph  the  fillet  must  be  drawn  in,  and  can  be 


most  easily  determined  by  making  a  mating 
tooth  of  paper,  and  trying  it  in  several  posi- 
tions in  the  tooth  space,  as  in  the  figure. 

Except  on  gears  of  very  few  teeth  the 
strength  gained  will  not  warrant  the  trouble 
of  constructing  the  fillet. 


45. — THE    EQUIDISTANT    SERIES. 


When  arranging  an  odontograph  for 
drafting  teeth,  or  a  set  of  cutters  for  cutting 
them,  we  must  make  one  sizing  value  do 
duty  for  an  interval  of  several  teeth,  for  it  is 
impracticable  to  use  different  values  for  two 
or  three  hundred  different  numbers  of  teeth. 
The  object  of  the  equidistant  series  is  to  so 
place  these  intervals  that  the  necessary  errors 
are  evenly  distributed,  each  sizing  value 
being  made  to  do  duty  for  several  numbers 
each  way  from  the  number  to  which  it  is 
fitted,  and  being  no  more  inaccurate  than  any 
other  for  the  extreme  numbers  that  it  is 
forced  to  cover. 

This  series  is  readily  computed  for  any 
case  that  may  arise,  and  with  a  degree  of  ac- 
curacy that  is  well  within  the  requirements 
^>f  practice;  by  the  formula 


.    as 
,      *-.  +  - 

in  which  a  is  the  first  and  z  is  the  last  tooth 
of  the  interchangeable  series  to  be  covered; 
n  is  the  number  of  intervals  in  the  series,  and 
*  is  the  number  in  the  series  of  any  interval 
of  which  the  last  tooth  t  is  required. 

For  example,  it  is  required  to  compute  the 
series  here  used  for  the  cycloidal  odonto- 
graph, having  twelve  tabular  numbers  to 
cover  from  twelve  teeth  to  a  rack. 

Putting  a  =  12,  z  =  infinity,  and  n  =  12, 
the  formula  becomes 

12  X  12  12  X  12  144 


t  = 


12  — s-f-0  ~  12  — 


and  then,  by  putting  «  successively  equal  to 
1,  2,  3,  4,  5,  6,  7,  8,  9,  10,  11  and  12,  we  get 
the  series  of  last  teeth,  13r'T,  14|,  16,  18,  20|, 
24,  28f,  36,  48,  72,  144,  and  infinity.  These 
give  the  required  equidistant  series  of  inter- 
vals. 


12 

13  to  14, 
15  to  16, 
17  to  18, 
19  to  21, 
22  to  24, 


25  to  29 
30  to  36, 
37  to  48, 
49  to  72, 
73  to  144, 
145  to  a  rack  ; 

and  the  method  is  as  easily  applied  to  any 
other  practical  example. 

This  formula  and  method  is  independent  of 
the  form  and  of  the  length  of  the  tooth,  and 
therefore  is  applicable  to  all  systems  under 
all  circumstances.  This  is  proper  and  con- 
venient, for  these  elements  can  be  eliminated 
without  vitiating  the  results  or  destroying  the 
"equidistant"  characteristic  of  the  series. 
The  formula  is  an  approximation  based  upon 
an  assumption,  but  nothing  more  convenient 
or  more  accurate  has  so  far  been  devised  by 
laboriously  considering  all  the  petty  elements 
involved. 

The  sizing  value,  or  number  for  which  the 
tabular  number  is  computed,  or  the  cutter  is 
accurately  shaped,  can  best  be  placed,  not  at 
the  center  of  the  interval,  but  by  considering 
the  interval  as  a  small  series  of  two  intervals, 
and  adopting  the  intermediate  value.  The 
sizing  value  for  the  interval  from  c  to  d  is 

given  by  the  formula 

2    cd 


Thus,  the  sizing  value  for  the  interval 
from  37  to  48  teeth  should  be  41.8,  and  that 
for  the  interval  from  145  to  a  rack  should  be 
290. 

It  is  sometimes  the  practice  to  size  the  cut- 
ter for  the  lowest  number  in  its  interval,  on 
the  ground  that  a  tooth  that  is  considerably 
too  much  curved  is  better  than  one  that  is 
even  a  little  too  flat.  This  makes  the  last 
tooth  of  the  interval  much  more  inaccurate 
than  if  the  medium  number  was  used. 


Friction  of  Approach. 


46. — THE  HUNTING  COG. 


It  is  customary  to  make  a  pair  of  cast  gears 
with  incommensurable  numbers  of  teeth  so 
that  each  tooth  of  each  gear  will  work  with 
all  the  teeth  of  the  other  gear.  If  a  pair  of 
equal  gears  have  twenty  teeth  each,  each 
tooth  will  work  with  the  same  mating  tooth 
all  the  time;  but  if  one  gear  has  twenty  and 
the  other  twenty-one  teeth,  or  any  two  num- 
bers not  having  a  common  divisor,  each  tooth 
will  work  with  all  the  mating  teeth  one  after 
the  other. 

The  object  is  to  secure  an  even  wearing 
action;  each  tooth  will  have  to  work  with 
many  other  teeth,  and  the  supposition  is  that 


all  the  teeth  will  eventually  and  mysteriously 
be  worn  to  some  indefinite  but  true  shape. 

It  would  seem  to  be  the  better  practice  to 
have  each  tooth  work  with  as  few  teeth  as 
possible,  for  if  it  is  out  of  shape  it  will  dam- 
age all  teeth  that  it  works  with,  and  the 
damage  should  be  confined  within  as  narrow 
limits  as  possible.  If  a  bad  tooth  works  with 
a  good  one  it  will  ruin  it,  and  if  it  works 
with  a  dozen  it  will  ruin  all  of  them.  It  is 
the  better  plan  to  have  all  the  teeth  as  near 
perfect  as  possible,  and  to  correct  all  evident 
imperfections  as  soon  as  discovered. 


47. — THE   MORTISE    WHEEL. 

Another  venerable  relic  of  the  last  century 
is  the  "mortise"  gear,  Fig.  32,  having 
wooden  teeth  set  in  a  cored  rim,  in  which 
they  are  driven  and  keyed. 

Where  a  gear  is  subjected  to  sudden  strains 
and  great  shocks,  the  mortise  wheel  is  better, 
and  works  with  less  noise  than  a  poor  cast 
gear,  and  will  carry  as  much  as  or  more 
power  at  a  high  speed  with  a  greater  dura- 
bility. But  in  no  case  is  it  the  equal  of  a 
properly  cut  gear,  while  its  cost  is  about  as 
great. 

In  times  when  large  gears  could  not  be  cut, 
and  when  the  cast  tooth  was  not  even  ap- 
proximately of  the  proper  shape,  the  mortise 
wheel  had  its  place,  but  now  that  the  large 
cut  gear  can  be  obtained  the  mortise  gear 
should  be  dropped  and  forgotten. 


Mortise  wheel 

Fig.  32. 


48. 


FRICTION  OF  APPROACH. 


When  the  point  of  action  between  two 
teeth  is  approaching  the  pitch  point,  that  is, 
when  the  action  is  approaching,  the  friction 
between  the  two  tooth  surfaces  is  greater  than 
when  the  action  is  receding.  This  extra  fric- 
tion is  always  present,  but  is  most  trouble- 
some when  the  surfaces  are  very  rough,  as  on 
cast  teeth,  giving  little  trouble  when  the  teeth 
are  properly  shaped  and  well  cut.  When  the 
roller  pin  gear  (93)  is  used,  the  friction 
between  the  teeth  is  rolling  friction,  and  is 
no  greater  on  the  approach  than  on  the  recess. 


tion  of 
roach 


rig.  33. 


22 


Efficiency. 


The  difference  in  the  friction  is  probably 
due  to  the  difference  in  the  direction  of  the 
pressure  between  the  small  inequalities  to 
which  all  friction  is  due.  When  the  gear  D, 
Fig.  33,  is  the  driver,  the  action  between  the 
teeth  is  receding,  and  the  inequalities  lift  over 
each  other  easily,  while  if  F  is  the  driver, 
the  action  is  approaching,  and  the  inequalities 
tend  to  jam  together. 

In  the  exaggerated  case  illustrated,  it  is  plain 
that  the  teeth  are  so  locked  together  that  ap- 
proaching action  is  impossible,  while  it  is 
equally  plain  that  motion  in  the  other  direc- 
tion is  easy.  The  same  action  takes  place  in 
a  lesser  degree  with  the  small  inequalities  of 
ordinary  rough  surfaces. 


The  action  of  the  common  friction  pawlr 
which  works  freely  in  one  direction  and  jam& 
hard  in  the  other,  is  upon  the  same  principle. 
A  weight  may  be  easily  dragged  over  a  rough 
surface  that  it  could  not  be  pushed  over  by  a 
force  that  is  not  parallel  to  the  surface. 

The  extra  friction  of  approaching  action 
can  be  avoided  by  giving  the  driver  the  long- 
est face.  When  the  driver  has  faces  only, 
and  the  follower  has  only  flanks,  the  action  is 
particularly  smooth. 

Teeth  that  are  subject  to  excessive  maxi- 
mum obliquity,  such  as  cycloidal  teeth,  should 
not  be  selected  for  rough  cast  gearing,  for  it 
is  the  maximum  rather  than  the  average  obli- 
quity that  has  the  greatest  influence. 


49. — EFFICIENCY  OF   GEAR  TEETH. 


Much  has  been  written,  but  very  little  has 
been  done  to  determine  the  efficiency  of  the 
teeth  of  gearing  in  the  transmission  of  power, 
and  therefore  but  little  of  a  definite  nature 
can  be  said.  The  question  is  mostly  a  prac- 
tical one,  and  should  be  settled  by  experi- 
ment rather  than  by  analysis. 

The  only  known  experiments  upon  the  fric- 
tion of  spur  gear  teeth  are  the  Sellers  experi- 
ments, more  fully  detailed  in  (112),  and  but. 
one  of  these  relates  to  the  spur  gear.  From 
that  one  it  is  known  that  a  gear  of  twelve 
teeth,  two  pitch,  working  in  a  gear  of  thirty- 
nine  teeth,  has  an  efficiency  varying  from 
ninety  per  centum  at  a  slow  speed  to  ninety- 
nine  per  centum  at  a  high  speed.  That  is, 
an  average  of  five  per  centum  of  the  power 
received  is  wasted  by  friction  at  the  teeth  and 
shaft  bearings.  This  result  is  probably  a 
close  approximation  to  that  for  any  ordinary 
practical  case. 

Although  theory  can  do  nothing  to  de- 
cide such  a  question  as  this,  it  can  do  much 
to  indicate  probable  results.  • 

If  a  pair  of  involute  teeth,  for  example, 
move  over  a  certain  distance,  w,  either  way 
from  the  pitch  point,  the  distance  being  mea- 
sured on  the  pitch  line,  they  will  do  work  that 
is  theoretically  determined  by  the  formula  : 

/  P        k  ±  h 
work    done  =  •—-     .     -= — =-    w9 

&  K    fl 

in  which  /  is  the  coeflBcient  of  friction,  P  is 


the  pressure,  and  k  and  h  are  the  pitch  radii 
of  the  gears.  The  positive  sign  is  to  be  used 
for  gears  in  external,  and  the  negative  sign, 
for  those  in  internal  contact. 

The  loss  by  friction,  as  shown  by  the  for- 
mula, decreases  directly  as  the  diameters  in- 
crease, the  proportion  of  the  diameters  being 
constant. 

The  loss  increases  rapidly  with  the  distance 
of  the  point  of  action  from  the  pitch  point 
When  the  contact  is  at  the  pitch  point  the 
teeth  do  not  slide  on  each  other,  and  there  is 
no  loss,  but  away  from  that  point  the  loss  is 
as  the  square  of  the  distance  in  this  case,  and 
in  a  still  greater  proportion  in  the  case  of  the 
cycloidal  tooth.  Therefore  a  short  arc  of 
action  tends  to  improve  the  efficiency. 

It  has  been  satisfactorily  determined  that 
the  loss  is  greater  during  the  approaching 
than  during  the  receding  action.  This  is  not 
shown  by  the  formula,  but  it  may  be  laid  to 
a  variation  in  the  coefficient  /. 

The  formula  shows  that  the  loss  is  inde- 
pendent of  the  width  or  face  of  the  gear, 
and  therefore  strength  can  be  increased  by 
widening  the  face,  without  increasing  the 
friction. 

If  the  work  of  internal  gearing  is  com- 
pared with  that  of  external  gearing  of  the 
same  sizes,  the  losses  are  in  the  proportion, 
k—  h 
k-\-h' 


Strength. 


23 


BO  that  the  internal  gear  is  much  the  more 
economical,  particularly  when  the  gear  and 
pinion  are  nearly  of  the  same  size.  If  the 
gear  is  twice  the  size  of  the  pinion  the  loss 
is  but  one-third  of  the  loss  when  both  gears 
are  external. 

Small  improvement  can  DC  effected,  by  put- 
ting a  small  pinion  inside  rather  than  outside 
of  a  large  gear.  A  six-inch  pinion  working 
with  a  six-foot  gear  has  but  1.18  times  the 
loss  by  the  same  gears,  when  the  gear  is  in- 
ternal. 

Theoretical  efficiency  is  discussed  at  great 
length  in  the  Journal  of  the  Franklin  Insti- 
tute, for  May,  1887:  Also  by  Reuleaux,  and 
again  by  Lanza,  in  the  Transactions  of  the 


American  Society  of  Mechanical  Engineers 
for  1887,  and  the  discussion  has  been  carried 
far  enough. 

A  series  of  experiments  with  gear  teeth  oi 
various  sizes  and  forms,  of  various  metals, 
would  add  greatly  to  our  knowledge  of  this 
important  matter. 

A  true  determination  of  the  efficiency  of 
the  rough  cast  gear,  as  compared  with  that  of 
the  cut  gear,  would  tend  to  discourage  the 
use  of  the  former  for  the  transmission  of 
power,  for  experiment  would  undoubtedly 
show  that  the  power  wasted  by  the  cast  gear 
would  soon  pay  the  difference  in  cost  of  the 
better  article. 


50. — STRENGTH  OF    A  TOOTH. 


The  strength  of  a  tooth  is  the  still  load  it  will 
carry,  suspended  from  its  point,  and  is  to  be 
carefully  distinguished  from  the  horse-power, 
or  the  load  the  gear  will  carry  in  motion. 

The  strength  of  a  substance  is  not  a  fixed 
element,  but  will  vary  with  different  samples, 
and  with  the  same  sample  under  different 
circumstances  ;  allowance  must  be  made  for 
the  amount  of  service  the  sample  has  seen, 
concealed  defects  must  be  provided  against, 
and  therefore  nothing  but  an  actual  test  will 
surely  determine  its  character. 

Although  no  possible  rule  can  be  depended 
upon,  the  ultimate  or  breaking  strength  of  a 
standard  cast-iron  tooth,  having  an  addendum 
about  equal  to  a  third  of  the  circular  pitch, 
will  average  about  three  thousand  five  hun- 
dred pounds  multiplied  by  the  face  of  the 
gear  and  again  by  the  circular  pitch,  both 
in  inches. 

But  a  tooth  should  never  be  forced  up  to 
its  ultimate  strength,  and  the  best  practice  is 
to  give  it  only  about  one-tenth  of  the  load  it 
might  possibly  bear,  so  that  the  following 
rule  should  be  used  :  Multiply  three  hundred 
and  fifty  pounds  by  the  face  of  the  gear,  and 
again  by  the  circular  pitch,  both  in  inches, 
and  the  product  will  be  the  safe  working 
load  of  one  tooth. 

Example  :  A  cast-iron  gear  of  one  inch 
pitch,  and  two  inches  face,  will  safely  lift 
350  X  2  x  1  =  700  pounds,  although  it 
would  probably  lift  7,000  pounds. 


When  there  are  two  teeth  always  in  work- 
ing contact,  it  is  safe  to  allow  double  the 
load,  but  care  must  be  taken  that  both  teeth 
are  always  in  full  contact. 

A  hard  wood  mortised  cog  has  about  one- 
third  of  the  strength  of  a  cast-iron  tooth : 
steel  has  double  the  strength ;  wrought-iron 
is  not  quite  as  strong. 

A  small  pinion  generally  has  teeth  that  are 
weak  at  the  roots,  and  then  it  will  increase 
the  strength  to  shroud  the  gear  up  to  its 
pitch  line,  but  shrouding  will  not  strengthen 
a  tooth  that  spreads  towards  its  base,  like  an 
involute  tooth,  and  when  the  face  of  the 
gear  is  wide  compared  with  the  length  of  the 
tooth  the  shroud  is  of  /ittle  assistance. 

It  does  not  increase  the  strength  of  a  tooth 
to  double  its  pitch,  for  when  the  pitch  is 
increased  the  length  is  also  increased,  and  the 
strength  is  still  in  direct  proportion  to  the 
circular  pitch,  wnile  the  increase  has  reduced 
the  number  of  teeth  fu  contact  at  a  time. 

Cut  gears  and  cast  gears  are  about  equal 
as  to  actual  strength,  with  the  advantages  in 
favor  of  the  cut  gear,  that  hidden  d  3fects  are 
likely  to  be  discovered,  and  that  it  is  not  as 
liable  to  undue  strains  on  account  of  defective 
shape. 

The  rules  for  strength  must  not  be  used  for 
gears  running  at  any  considerable  speed,  for 
they  are  intended  only  for  slow  service,  as  in 
cranes,  heavy  elevators,  power  punches,  etc, 


J-f or se- Power. 


f 


51. — HORSE-POWER   OP   CAST  GEARS. 


The  horse-po\ver  of  a  gear  is  the  amount 
of  power  it  may  be  depended  upon  to  carry 
in  continual  service. 

It  is  very  well  settled  that  continual  strains 
and  impact  will  change  the  nature  of  the 
metal,  rendering  it  more  brittle,  so  that  a 
tooth  that  is  perfectly  reliable  when  new 
may  be  worthless  when  it  has  seen  some  years 
of  service.  This  cause  of  deterioration  is 
particularly  potent  in  the  case  of  rough  cast 
teeth,  for  they  can  only  approximate  to 
true  shape  required  to  transmit  a  uniform 
speed,  and  the  continual  impact  from  shocks 
and  rapid  variations  in  the  power  carried 
must  and  does  destroy  the  strength  of  the 
metal. 

There  are  about  as  many  rules  for  com- 
puting the  power  of  a  gear  as  there  are 
manufacturers  of  gears,  each  foundryman 
having  a  rule,  the  only  good  one,  which  he 
has  found  in  some  book,  and  with  which  he 
will  figure  the  power  down  to  so  many 
horses  and  hundredths  of  a  horse  as  con- 
fidently as  he  will  count  the  teeth  or  weigh 
the  casting. 

Even  among  the  standard  writers  on  en- 
gineering subjects  the  agreement  is  no  bet- 
ter, as  shown  by  Cooper's  collection  of 
twenty-four  rules  from  many  different  wri- 
ters, applied  to  the  single  case  of  a  five-foot 
gear.  See  the  "Journal  of  the  Franklin 
Institute"  for  July,  1879.  For  the  single 
case  over  twenty  different  results  were  ob- 
tained, ranging  from  forty-six  to  three- 
hundred  horse-power,  and  proving  conclu- 
sively that  the  exact  object  sought  is  not  to  be 
obtained  by  calculation. 

This  variety  is  very  convenient,  for  it  is 
always  possible  to  fit  a  desired  power  to 
a  given  gear,  and  if  a  badly  designed  gear 
should  break,  it  is  a  simple  matter  to  find  a 
rule  to  prove  that  it  was  just  right,  and  must 
have  met  with  some  accident. 


Although  no  rule  can  be  called  reliable, 
the  one  that  appears  to  be  the  best  is  that 
given  by  Box,  in  his  Treatise  on  Mill  Gear- 
ing. Box's  rule,  which  is  based  on  many 
actual  cases,  and  which  gives  among  the 
lowest,  and  therefore  the  safest  results,  is  by 
the  formula: 

„  12  c2/  *J~dn 

Horse-power  of  a  cast  gear  =  THA^ — 

J  ,UUO 

in  which  c  is  the  circular  pitch,  /is  the  face, 
the  M  is  the  diameter,  all  in  inches,  and  n  is  the 
number  of  revolutions  per  minute. 

Example  :  A  gear  of  two  feet  diameter, 
four  inches  face,  two  inches  pitch,  running 
at  one  hundred  revolutions  per  minute,  will 
transmit 


12  X  2  X  2:x  4  X  A/  24  x  100 


1,000 


=  9.4  h.  p. 


For  bevel  gears,  take  the  diameter  and 
pitch  at  the  middle  of  the  face. 

It  is  perfectly  allowable,  although  it  is  not 
good  practice,  to  depend  upon  the  gear  for 
from  three  to  six  times-the  calculated  power, 
if  it  is  new,  well  made,  and  runs  without 
being  subjected*  to  sudden  shocks  and  varia- 
tions of  load. 

The  influence  of  impact  and  continued 
service  will  be  appreciated  when  it  is  con- 
sidered that  the  gear  in  the  example,  which 
will  carry  9.4  horse-power,  will  carry  seventy 
horse-power  if  impact  is  ignored,  and  the 
ultimate  strength  of  the  metal  is  the  only 
dependence. 

A  mortise  gear,  with  wooden  cogs,  will 
carry  as  much  as,  or  more  than  a  rough  cast- 
iron  gear  will  carry,  although  its  strength  is 
much  inferior.  The  elasticity  of  the  wood 
allows  it  to  spring  and  stand  a  shock 
that  would  break  a  more  brittle  tooth  of 
much  greater  strength.  And,  for  the  same 
reason,  a  gear  will  last  longer  in  a  yielding 
wooden  frame  than  it  will  in  a  rigid  iron 
frame. 


52. — HORSE-POWER  OP   CUT   GEARS. 


We  know  a  little,  and  have  to  guess  the 
rest,  as  to  the  power  of  a  cast  gear,  but  with 
respect  to  that  of  a  cut  gear  we  are  not  as 
well  posted,  for  there  are  no  experimental 


data   upon    which  a  reliable    rule    can    be 
founded. 

Admitting,   as  we  must,   that    impact    is 
the  chief  cause  of  the  deterioration  of  the 


The  Involute    Tooth. 


cast  gear,  we  are  at  liberty  to  assume  that  a 
properly  cut  and  smoothly  running  cut  gear 

ris  much  more  reliable. 

No  definite  rule  is  possible,  but  we  can 
safely  assume  that  a  cut  gear  will  carry  at 
least  three  times  as  much  power  as  can  be 
trusted  to  a  cast  gear  of  the  same  size. 
i  The  great  reliance  of  those  who  claim  that 
a  cast  gear  is  superior  to  a  cut  gear  is  upon 

,  the  hard  scale  with  which  the  cast  tooth  is 
covered.  This  scale  is  not  over  one-hun- 
dredth of  an  inch  thick,  is  rapidly  worn 


away,  and  is  of  no  account  whatever.    From 
that  point  of  view  it  is  difficult  to  explain 
why  a  wooden  tooth  will  outwear  an  iron  one, 
I  although  it  is  softer  than  the  softest  cut  iron. 
Assuming  that  a  cut  gear  is  about  three 
'  times  as  reliable  as  a  cast  gear,  we  can  com- 
pute its  power  by  the  formula  : 

d  n 


c'f 
I  Horse-power  of   a  cut  gear  =  - 

in  which  c  is  the  circular  pitch,  /  is  the  face, 
and  d  is  the  pitch  diameter,  all  in  inches,  and 
n  is  the  number  of  revolutions  per  minute. 


3.  THE    IKVOLUTE    SYSTEM. 


53. — THE  INVOLUTE    TOOTH. 

The  simplest  and  best  tooth  curve,  theo- 
retic-ally, as  well  as  the  one  in  greatest  prac- 
tical use  for  cut  gearing,  is  the  involute. 

The  involute  tooth  system  is  based  on  the 

'  straight  rack  odontoid,  (31)  and  Fig.  21,  and 

\  it  is  illustrated  by  Fig.  34.     If  the  four  odon- 

!  toids  of  the  rack  outline  are  equally  inclined 

i  to  the  pitch  line,  the  resulting  tooth  system 

will  be  completely  interchangeable;  but  if, 

as  in  Fig.  35,  the  face  and  flank  are  inclined 

at  different  angles  of  obliquity,  T  S  K  and 

T  S  K',  the  system  is  not  interchangeable, 

although  otherwise  perfect. 

The  rack  odontoid  cannot  have  a  corner  or 
change  of  direction  anywhere  except  at  the 
pitch  line,  without  causing  a  break  in  the 
line  of  action. 

As  the  normals  p  q  are  parallel,  the  line  of 
action  is  a  straight  line  W  O  W  at  right 
angles  to  the  rack  odontoid.  The  inter- 
changeable line  of  action  is  continued  in  a 
straight  line  on  both  sides  of  the  pitch  line, 
bui  the  non-interchangeable  line  changes  di- 
rection at  that  line. 

In  accordance  with  the  universal  custom 
we  will  consider  that  the  involute  tooth  is 
'always  interchangeable,  having  a  single  angle 
ot  obliquity. 


Z/ic  involute  tooth 
interchangeable 


Fiff.  34. 


26 


Involute  Interference. 


54. — THE  CUSP. 


As  a  circle  t  c,  Fig.  34,  can  always  be  drawn 
tangent  to  the  line  of  action  at  an  interfer- 
ence point  i,  from  the  center  b  of  any  pitch  line 
B,  there  will  always  be  a  cusp  in  the  curve  at 
the  point  c  (16),  and  at  that  point  the  working 
part  of  the  curve  must  stop.  The  working 
part  of  the  rack  tooth  must  end  at  the  limit 
line  i  L  through  the  interference  point  i. 

The  working  curves  of  any  two  teeth  that 
work  with  each  other  must  each  end  at  the 
line  drawn  through  the  interference  point  of 
the  other,  Fig.  43,  being  limited  by  limit 
lines  1 1  and  L  L. 


r  The  second  branch  c  m'  of  the  curve  is 
equal  to  the  first  branch  c  m,  but  is  re- 
versed in  direction.  The  second  cusp  is  at 
infinity,  and  therefore  has  no  practical  ex- 
istence. 

The  tangent  circle  i  c,  through  the  inter- 
ference point  and  the  cusp,  is  called  the 
"base  line." 

It  is  customary  to  continue  the  flank  of  the 
tooth  inside  the  base  line  by  a  straight  radial 
line,  as  far  as  may  be  necessary  to  allow  the 
mating  gear  to  pass. 


55. — INTERFERENCE. 

When  the  point  of  the  tooth  is  continued 
beyond  the  limit  line  it  will  interfere  with  and 
cut  away  a  portion  of  the  working  curve  of 
the  mating  tooth.  Fig.  36  shows  a  rack  tooth 
working  with  the  tooth  of  a  small  pinion,  and 
cutting  out  its  working  curve. 

This  cut  is  not  confined  to  the  flank,  but 
extends  across  the  pitch  line  into  the  face,  as 
shown  by  the  line  qmn.  The  rack  tooth  of 
the  figure  will  not  work  with  the  pinion  tooth 
unless  it  is  cut  off  at  the  limit  line  1 1  through 
the  interference  point  i. 

The  mathematical  action  still  continues, 
and  the  figure  shows  the  rack  tooth  in. action 
at  k  with  the  second  branch  of  the  curve. 


Effect  of  Interference 

.  36. 


56. — ADJUSTABILITY. 


An  interesting  and  in  many  cases  a  valua- 
ble feature  of  the  involute  curve,  and  one 
that  is  confined  to  it,  is  the  fact  that  its  posi- 
tion as  a  whole  with  regard  to  the  mating 
curve  is  adjustable. 

Two  involutes,  each  with  its  base  line,  will 
work  together  in  perfect  tooth  contact  when 
they  are  moved  with  respect  to  each  other, 
as  long  as  they  touch  at  all.  The  lines  of 
action  and  the  pitch  lines  will  shift  as  the 
curves  are  moved,  and  will  accommodate 
themselves  to  the  varying  position  of  the 
base  lines. 

But  this  valuable  feature  of  the  involute 
curve  is  not  always  available,  and  involute 
gears  are  not,  as  commonly  supposed,  neces- 


sarily adjustable,  for  the  conditions  are  often 
such  that  the  teeth  will  fail  to  act  when  the 
centers  are  moved,  except  within  very  narrow 
limits.  Care  must  be  taken  that  the  arc  of 
action  is  not  so  reduced  by  separating  the 
centers  of  the  gears  that  it  is  less  than  the  cir- 
cular pitch,  for  the  former  arc  is  variable  and 
the  latter  is  fixed.  Care  must  also  be  taken 
that  the  working  curve  is  not  pushed  over  the 
limit  line  when  the  centers  are  drawn  to- 
gether. 

In  any  limiting  case,  such  as  in  Fig.  43,  the 
centers  are  not  adjustable.  The  gears  of  the 
standard  set  are  either  not  adjustable  at  all 
or  are  so  within  very  narrow  limits,  on  ac- 
count of  the  correction  for  interference. 


Involute    Construction. 


27 


57.— CONSTRUCTING  THE  INYOLUTE  BY  POINTS. 


The  simple  involute  curve  can  be  con- 
structed by  points  by  the  general  method  of 
(24),  but  it  is  much  better  to  take  advantage 
of  the  property  that  it  is  an  involute  of  its 
base  circle,  and  construct  it  by  the  rectifica- 
tion of  that  circle. 

As  in  Fig.  37  any  convenient  small  dis- 
tance A  G  is  taken  on  the  dividers,  and  the 
points  on  the  curve  located  by  stepping 
along  the  circle  and  its  tangent  from  any 
given  point  to  any  desired  point. 

This  method  is  so  accurate,  if  care  is  taken 
to  step  accurately  on  the  line,  that  the  curve 
seldom  needs  correction;  but,  when  great  ac- 
curacy is  required,  correction  can  be  applied 
at  the  rate  of  one- thousandth  of  an  inch  to 
the  step,  if  the  length  of  the  step  is  regulated 
by  the  diameter  of  the  circle  according  to  the 
following  table: 

Diameter  of  Circle : 

12345       678       9       10     11     12 
Length  of  Step  : 
.17    .26    .37    .46    .53    .60    .67    .73    .76     .79    .82    .84 

For  example:    If  the  circle  of  Fig.  37  is 


Construction 
points 

Fig.  37. 

four  inches  in  diameter,  and  the  dividers  are 
set  to  .46  inch,  the  true  curve,  A  b'  d',  will  be 
outside  of  the  constructed  curve  A  b  d  by  .002 
inch  at  b  and  .005  inch  at  d. 

From  the  table  we  can  form  the  handy  and 
sufficiently  accurate  rule  that  the  length  of 
the  slep  should  be  about  one-tenth  of  the  di- 
ameter of  the  circle,  for  a  correction  of  about 
one-thousandth  of  an  inch  per  step. 

Having  thus  found  several  points  of  the  in- 
volute, we  can  draw  it  in  by  hand,  or  by  con- 
structing a  template,  or  by  finding  centers 
from  which  approximately  accurate  circular 
arcs  can  be  drawn. 


58. — THE  STANDARD  INVOLUTE  TOOTH. 


The  tooth  that  is  selected  for  general  use, 
and  the  one  that  is  the  best  for  all  except  a 
few  special  cases  and  limiting  cases,  is  the  in- 
terchangeable tooth  having  an  angle  of  ob- 
liquity of  fifteen  degrees,  an  addendum  of 
one-third  the  circular  pitch,  or  one  divided  by 
the  diametral  pitch,  and  a  clearance  of  one- 
-eighth  of  the  addendum. 

The  standard  to  which  involute  cutters  are 
made  is  slightly  different,  having  an  angle  of 
14°  28'  40",  the  sine  of  which  is  one-quarter, 
.and  a  clearance  of  one-twentieth  of  the  circu- 
lar pitch. 

If  the  obliquity  is  15°  the  smallest  possible 
pair  of  equal  gears  have  11.72  teeth,  and 


therefore  12  is  the  smallest  gear  of  the  inter- 
changeable set. 

The  base  distance,  the  distance  of  the  base 
line  inside  of  the  pitch  line,  is  about  one-tifty- 
ninth  of  the  pitch  diameter,  and  one-sixtieth 
is  a  convenient  fraction  for  practical  use. 

The  limit  points  of  the  whole  set  must  be 
determined  by  that  of  the  twelve-toothed 
gear,  for  any  gear  of  the  set  may  be  required 
to  work  with  that  one,  and  the  working  curve 
of  each  tooth  must  end  at  the  point  thus  de- 
termined. As  the  limit  point  is  always  in- 
side of  the  addendum  line  there  must  always 
be  a  false  extension  on  the  tooth,  the  point 
being  rounded  over  outside  of  the  limit  point. 


59. — THE  INVOLUTE    ODONTOGRAPH. 


As  the  base  line  must  always  be  drawn,  it 
is  advisable,  to  save  work,  to  locate  the  cen- 
ters of  the  approximate  circular  arcs  upon 
that  line.  It  is  also  necessary  that  the  points 
-of  the  teeth  shall  be  rounded  over,  to  avoid 


interference.    These    requirements   made  it 
impracticable  to  compute  the  positions  of  the 
centers,  and  an  empirical  rule  had  to  be  adopt- 
ed instead. 
Teeth  were  carefully  drawn  by  the  stepping 


Ten  arid  Eleven  Involute    Teetli. 


method  of  (57)  on  a  very  large  scale,  one- 
quarter  pitch,  giving  a  tooth  eight  inches  in 
length.  These  teeth  were  corrected  for  inter- 
ference by  giving  them  epicycloidal  points 
that  would  clear  the  radial  flanks  of  the 
twelve-toothed  pinion. 

Then  the  proper  centers  on  the  base  line 
were  determined  by  repeated  trials,  and  tooth 
curves  obtained  that  would  agree  with  the 
true  involute  up  to  the  limit  point,  and  still 


clear  the  corrected  point.  The  odontograph 
table  is  a  record  of  these  radii,  which  are  be- 
lieved to  be  as  nearly  coyrect  as  the  given 
:  conditions  will  permit. 

It  was  found  that  separate  curves  were, 
required  for  face  and  flank  up  to  thirty-six 
teeth,  but  that  one  curve  would  answer  for 
teeth  beyond. 

It  was  found  necessary  to  devise  a  separate 
method  for  drafting  the  rack  tooth. 


60. — TEN  AND  ELEVEN   TEETH. 


Theoretically  the  twelve-toothed  pinion  is 
the  smallest  standard  gear  that  will  have  an 
arc  of  action  as  great  as  the  circular  pitch, 
but  ten  and  eleven  teeth  may  be  used  with 
an  error  that  is  not  practically  noticeable. 
Fig.  38  shows  a  pair  of  ten-toothed  gears  in 


action.  They  can  be  in  correct  action  only 
when  the  point  of  contact  is  between  the  two 
interference  points  i  and  J,  but  they  will  be 
in  practical  contact  for  a  greater  and  suffi- 
cient distance 


Fig.  3S. 


OdontoyrapJiic  pair 


61.— A  BAD   RULE. 


There  is  a  simple  and  worthless  rule  for 
involute  teeth  that  deserves  notice  only  be- 
cause it  is  considerably  in  use. 

It  constructs  the  whole  tooth  curve,  face  and 
flank,  for  all  numbers  of  leeth,  as  a  single 


arc  from  a  center  on  the  base  line,  and  with  a 
radius  equal  to  one-quarter  of  the  pitch  radius, 
Fig.  39. 

This  is  wonderfully  convenient,  but  the 
convenience  is  purchased  at  the  expense  of 


The  Involute    Odontograph. 


29 


'ordinary  accuracy,  for  the  rule  is  not  even 
approximately  correct.  It  is  handy,  and 
nothing  else. 

Figs.  38  and  40  show  the  kind  of  teeth 
that  are  constructed  by  this  rule  on  gears  of 
ten  and  twelve  teeth,  where  its  error  is  the 
greatest,  and  it  is  reasonable  that  the  invo- 
lute tooth  should  not  be  in  great  favor  with 
those  who  have  been  taught  to  draw  it  thus. 

The  error  gradually  decreases,  until,  for 
more  than  thirty  teeth,  it  is  tolerably  correct, 
but  it  gives  the  rack  with  the  straight,  uncor- 
rected  working  face  that  would  interfere,  as 
shown  at  g,  Fig.  40. 

As  it  is  tolerable  only  for  thirty  or  more 


teeth,  and  not  good  then,  it  may  well  be 
dropped  altogether. 


bad  rule 


Figl  39. 


62. — USING  THE  INVOLUTE  ODONTOGRAPH. 

INVOLUTE    ODONTOGRAPH. 
STANDARD  INTERCHANGEABLE  TOOTH,  CENTERS  ON 

(For  Table  of  Pitch  Diameters  see  35.)       b     }  j 


Divide  by  the 
Diametral  Pitch. 

Multiply  by  the 
Circular  Pitch. 

Teeth. 

Face 

Flank 

Face 

Flank 

Radius. 

Radius. 

Radius. 

Radius. 

10 

2.28 

.69- 

.73 

.22 

11 

2.40 

.83 

.76 

.27 

12 

2  51 

.96 

.80 

.31 

13 

2.62 

.09 

.83 

.34 

14 

2.72 

22 

.87 

.39 

15 

2,82 

.34 

.90 

.43 

16 

2.92 

.46 

.93 

.47 

17 

3.02 

.58 

.96 

.50 

18 

3.12 

.69 

.99 

.54 

-19 

3.22 

.79 

1.03 

.57 

20 

3.32 

.89 

.06 

.60 

21 

3.41 

.98 

1  09 

.63 

22 

3.49 

2.06 

1  11 

.66 

23 

3.57 

2  15 

1.13 

.69 

24 

3.64 

2.24 

1.16 

.71 

25 

3.71 

2.33 

1  18 

.74 

26 

3.78 

2.42 

1  20 

.77 

27 

3.85 

2.50 

.23 

.80 

28 

3  92 

2.59 

25 

.82 

29 

3  99 

2.6T 

.27 

.85 

80 

4.06 

2.76 

.29 

.88 

81 

4.13 

2.85 

.31 

.91 

82 

4.20 

2.93 

.34 

.93 

33 

4  27. 

3  01 

36 

.96 

34 

4.33 

3  09 

.38 

.99 

35 

4  89 

3  16 

.39 

1.01 

36 

4.45 

3  23 

.41 

1.03 

87—40 

4.20 

1.34 

41-45 

4.63 

1.48 

46-51 

5  06 

1.61 

62-60 

5  74 

1.83 

61-70 

6  52 

2.07 

71-90 

7.72 

246 

91—120 

9.78 

3.11 

121-180 

13.38 

4.26 

-  r=  ^JL^A  l-o 

\  Fat,  f?^fJ(  .^)L_ 

30 


The  Involute    Odontograph. 


To  draft  the  tooth  lay  off  the  pitch,  ad- 
dendum, root,  and  clearance  lines,  and  space 
the  pitch  line  for  the  teeth,  as  in  Fig.  40. 

Draw  the  base  line  one-sixtieth  of  the  pitch 
diameter  inside  the  pitch  line. 

Take  the  tabular  face  radius  on  the  divid- 
ers, after  multiplying  or  dividing  it  as  re- 
quired by  the  table,  and  draw  in  all  the  faces 


from  the  pitch  line  to  the  addendum  line 
from  centers  on  the  base  line. 

Set  the  dividers  to  the  tabular  flank  radius, 
and  draw  in  all  the  flanks  from  the  pitch  line 
to  the  base  line. 

Draw  straight  radial  flanks  from  the  base 
line  to  the  root  line,  and  round  them  into  the 
clearance  line. 


Fig.  40. 


\. — SPECIAL   RULE   FOR   THE   RACK. 


Draw  the  sides  of  the  rack  tooth,  Fig.  40, 
as  straight  lines  inclined  to  the  line  of  centers 
c  0  c  at  an  angle  of  fifteen  degrees,  best 
found  by  quartering  the  angle  of  sixty  de- 
grees, 

Draw  the  outer  half  a  b  of  the  face,  one- 


quarter  of  the  whole  length  of  the  tooth, 
from  a  center  on  the  pitch  line,  and  with  a 
radius  of 

2.10  inches  divided  by  the  diametral  pitch. 
.67  inches  multiplied  by  the  circular  pitch. 


64. — DRAFTING  INTERNAL   GEARS. 


When  the  internal  gear  is  to  be  drawn,  the 
odontograph  should  be  used  as  if  the  gear 
was  an  ordinary  external  gear.  See  Fig.  41. 

But  care  must  be  taken  that  the  tooth  of 
the  gear  is  cut  off  at  the  limit  line  drawn 
through  the  interference  point  *  of  the  pin- 
ion. The  point  of  the  tooth  may  be  left  off 


altogether  or  rounded  over  to  get  the  appear- 
|  ance  of  a  long  tooth. 

The  pinion  tooth  need  not  be  carried  in  to 
the  usual  root  line,  but,  as  in  the  figure,  may 
just  clear  the  truncated  tooth  of  the  gear. 

The  curves  of  the  internal  tooth  and  of  its 
pinion  may  best  be  drawn  in  by  points  (57), 


The  Involute    Odontograph. 


31 


for  the  odomographic  corrected  tooth  is  not 
as  well  adapted  to  the  place  as  the  true  tooth, 
and  no  correction  for  interference  is  needed 
on  the  points  of  the  pinion  teeth  or  on  the 
flanks  of  those  of  the  gear. 


Care  must  be  taken  that  the  internal  teeth 
do  not  interfere  by  the  point  a  striking  the 
point  b,  as  they  will  if  the  pitch  diameters 
are  too  nearly  of  the  same  size. 


Internal  involutes 


65. — INVOLUTE   GEARS  FOR   GIVEN   OBLIQUITY  AND   ADDENDA. 


When  the  obliquity  and  addenda,  as  well 
as  the  pitch  diameter  and  number  of  teeth  in 
a  gear  are  given,  as  is  generally  the  case,  we 
can  proceed  to  draft  the  complete  gear  as 
follows: 

Draw  the  pitch  line  p  I,  Fig.  42,  the  ad- 
dendum line  a  I,  the  root  line  r  I,  and  the 
clearance  line  c  I,  as  given.  Draw  the  line  of 
action  I  a  at  the  given  obliquity  W  0  Z  =  K. 
Draw  the  base  line  b  I  tangent  to  the  line  of 
action.  Find  the  interference  point  *  by  bi- 
secting the  chord  0 t. 

Draw  the  involutes  i  a  m  and  t"  a"  in", 
and  a  a"  will  be  the  maximum  arc  of  ac- 
tion. 

If  the  given  arc  of  action  a  a'  is  not  great- 
er than  the  maximum  arc,  the  pitch  line  is 
to  be  spaced  and  the  tooth  curves  drawn  in 
from  the  base  line  to  the  addendum  line. 


These  tooth  curves,  when    small,  are    best 

drawn  as  circular  arcs  from  centers  on  or 

|  near  the  b#se  line,  one  center  x  for  the  flank 

from  the  base  line  to  the  pitch  line,  and 

another  center  y  for  the  face  from  the  pitch 

line  to  the  addendum  line.      One  involute 

i  a  m  should  be  carefully  constructed  by 

!  points,  and  then  the  required  centers  can  be 

1  found    by    trial.     One  center  and  arc  will 

!  often  answer  for  the  whole  curve,  and  it  is 

only  when  great  accuracy  is  required  that 

more  than  two  centers  will  be  necessary. 

Continue  the  flanks  of    the  teeth  toward 
center  by  straight  radial    lines,  and  round 
j  these  lines  into  the  clearance  line. 

If  the  interference  point  for  the  gear  that 

the  gear  being  drawn  is  to  work  with  is  at  I, 

i  within  the  addendum  line,  the  limit  line  1 1 

i  must  be  drawn  through  it,  and  the  points  of 


32 


Involute  Special   Cases. 


Fig.  42. 


Given  obliquity  and  addendum 


the  teeth  outside  of  this  limit  must  be  slightly 
rounded  over,  to  avoid  interference  (55). 
If  a  fillet  /  is  desirable,  to  strengthen  the 


tooth,  it  can  be  drawn  in  by  the  method 

of  (44). 


66. — INVOLUTE  GEARS  FOR  GIVEN  NUM-  ' 
BERS  OF  TEETH. 

When  the  numbers  of  teeth  and  the  pitch 
lines  are  the  only  given  details,  the  shape  and 
action  of  the  tooth  depends  upon  the  obli- 
quity, and  the  action  will  fail  if  the  angle  is 
too  small.  The  principal  object  is  to  deter- 
mine the  least  possible  angle  that  is  permitted 
by  the  given  pitch  diameters  and  numbers  of 
teeth. 

Draw  the  pitch  lines  P  L  and  p  I,  Fig.  43, 
lay  off  the  given  pitch  arc,  as  a  straight  line 
c  d  or  C  D,  at  right  angles  to  the  line  of 
centers,  and  draw  the  line  C  d  or  c  D.  Then 
the  required  line  pf  action  will  be  I  a  pass- 
ing through  0  at  right  angles  to  c  D  or  G  d. 
The  complete  teeth  can  then  be  drawn  in  as 
previously  directed. 

In  this  case,  the  obliquity  WO  Z being  the 
least  possible,  the  limit  lines  and  the  adden- 
dum lines  must  coincide,  but  the  addenda 
may  be  reduced  by  increasing  the  angle. 


Fig.  43. 


la 


Given  numbers  of  teeth 


Limiting  Involute    Teeth. 


33 


67. — INVOLUTE  GEARS  FOR  GIVEN  OBLIQUITY. 


When  the  pitch  diameters  and  the  obliquity 
are  the  only  given  details,  the  lines  C I  and 
c  i,  Fig.  43,  drawn  from  the  centers  at  right 
angles  to  the  line  of  action,  will  determine 
the  limit  lines.  The  maximum  arc  of  action 
a  a'  may  be  found  either  by  drawing  the 
involutes  i  a  and  la',  or  by  continuing  the 
line  G  I  to  the  line  c  d,  and  measuring  the 


required  distance  c  d.    Any  arc  of  action  less 
than  a  a'  may  be  used. 

The  drawings  should  always  be  made  to 
a  scale  of  one  tooth  to  the  inch  radius,  so 
that  the  pitch  arc  will  be  2*.  If  the  scale 
is  one  tooth  to  the  inch  of  diameter,  the 
pitch  arc  will  be  ?r. 


68.— INVOLUTE   GEARS  WITH  LESS   THAN  FIVE   EQUAL   TEETH. 


The  "method  of  Fig.  43  and  (66)  will  be 
found  to  apply  to  any  given  numbers  of 
teeth  not  less  than  five,  and  to  fail,  if  either 
gear  has  but  three  or  but  four  teeth.  Any 
external  gear  of  five  or  more  teeth  will  work 
with  any  external  gear  of  five  or  more  teeth, 
and  with  an  internal  gear  of  any  number  of 
teeth  unless  stopped  by  internal  interfer- 
ence (64). 

For  example,  if  a  pair  having  four  and  five 
teeth,  Fig.  44,  is  tried,  the  four-toothed 
pinion  will  fail,  because  its  tooth  will  come 
to  a  point  upon  the  line  of  action  before  it 
has  passed  over  the  required  pitch  arc.  The 
difficulty  cannot  be  remedied  by  increasing 
the  obliquity,  for  an  angle  that  would  allow 
the  four-toothed  pinion  to  act  would  also 
cause  the  five-toothed  pinion  to  fail. 

The  practical  limit  is  five  teeth,  but  the 
mathematical  limit  is  the  pair  having  the 
fractional  number  4. 62  teeth,  Fig.  45. 

The  four-toothed  pinion  will  not  work  with 
any  external  gear,  not  even  with  a  rack,  but 
it  will  work  with  an  internal  gear  that  has 
'  about  ten  thousand  teeth,  and  is  practically  a 
rack.  It  will  work  with  any  internal  gear 
having  less  than  ten  thousand  teeth,  and  Fig. 
46  shows  it  working  with  an  internal  gear  of 
six  teeth.  Internal  interference  will  prevent 
its  working  with  an  internal  gear  of  five 
teeth. 

The  three-toothed  pinion  has  no  practical 
action.  It  has  a  mathematical  action  with  in- 
ternal gears  of  3.56  or  less  teeth,  as  shown 
by  Fig.  47,  but  as  its  limit  is  less  than  four,  it 
cannot  work  with  any  whole  number.  The 
figure  shows  the  interference  at  a. 

The  extreme  mathematical  limit  may  be 
said  to  be  the  gear  of  2.70  teeth,  which  has  a 


theoretical  action  with  an  internal  gear  of  the 
same  size,  coinciding  with  it. 


4.69  X  4.62  limit  'for  equal  teeth 

Fig.  45. 

• 


Limiting  Involute    Teeth. 


Fig 


w 


69. — INVOLUTE  GEARS  WITH  LESS   THAN 
FIVE  UNEQUAL   TEETH. 

If  we  drop  the  condition  that  the  pitch 
line  must  be  equally  divided  into  tooth  and 
space  arcs,  we  can  make  gears  of  three  and  of 
four  teeth  work  with  external  gears  by  the 
method  of  (65).  The  failing  case  of  Fig.  44 
may  be  corrected  by  widening  the  failing 
tooth  until  it  acts,  and  narrowing  the  other 
tooth  to  correspond,  as  shown  in  broken  lines. 

In  this  way  a  four-toothed  pinion  will 
work  with  any  number  of  teeth  not  less  than 
5.57,  at  which  limit  both  gears  have  pointed 
teeth,  as  in  Fig.  48. 

The  three-toothed  pinion  will  work  with 
any  gear  having  10.17  or  more  teeth.  Fig.  49 
shows  the  3x10.17  limiting  pair,  and  Fig.  50 
shows  the  three-toothed  pinion  working 
with  an  internal  gear  of  five  teeth.  It  will 
not  work  with  an  internal  gear  of  four  teeth, 
on  account  of  internal  interference,  and  there- 
fore the  combination  shown  by  Fig.  50  may 
be  said  to  be  the  least  possible  symmetrical  in- 
volute pair. 

A  gear  of  2.70  teeth  will  work  with  a  rack, 
but  there  seems  to  be  no  way  to  make  a 
pinion  of  two  teeth  work  under  any  circum- 
stances. 


Fig.  47 


5.57  teeth 


4  teeth 

Fig.  48. 


Limiting  Involute    Teeth. 


35 


Fig.  49. 


3  X  10. 17 
Unequal  teetU 


Fig.  50 


70. — THE   MATHEMATICAL   LIMITS. 


The  above  results  for  low  numbered  pinions 
can  be  obtained  by  graphical  means,  but  that 
method  is  not  accurate  enough  to  determine 
the  limits  with  great  precision,  and  in  any 
case  is  tedious  and  laborious. 

The  mathematical  process  is  not  particu- 
larly difficult,  and  consists  in  repeated  trials 
with  given  formulae. 

To  determine  the  obliquity  at  which  a 
limiting  pinion  will  be  pointed  on  the  line  of 
action,  for  tooth  equal  to  space,  we  use  the 
formulae  : 

.  .  27T  M 

tan.  h  = 


M+n        90  ~ 

in  which  n  is  the  given  number  of  teeth  in 
the  pointed  gear,  Fig.  51,  M  is  the  number 
in  the  gear  having  the  radius  0  M,  and  h  is 
the  angle  0  c  I.  Knowing  n,  we  assume  a 
value  for  M,  and  from  that  find  a  value 
for  h  by  means  of  the  first  formula.  This 
value  of  7i,  tried  in  the  second  formula,  will 
give  an  error.  A  second  assumption  for  M 
will  give  a  second  error,  and  if  the  two 
errors  are  not  too  great  a  comparison  will 
nearly  locate  the  true  value  of  M. 

Knowing  n  and  M,  we  find  the  obliquity 
from 


tan.  K  =   — 


M+n 


pointed  pinion. 

Fig.  51. 

In  this  way  the  following  values  were  de- 
termined : 


n 

M 

K 

2.695 

1.26 

57°  49! 

3. 

1.51 

54°  20' 

4. 

2.86 

42°  29' 

4.62 

4.62 

34°  11' 

5. 

6.75 

28°  8' 

5.58 

00 

0 

Having  determined  the  obliquity  for  the 
pointed  pinion,  we  can  determine  the  least 
number  of  teeth  it  will  work  with  by  means 
of  the  following  formula  : 

180    n                     90 
Angle  B  = —  -=?  tan.  K ==  -4-  K 

IT        JV  -/V 

tan.  B  =  -==  tan.  K-\-  tan.  K 
in  which  N  is  the  required  least  number. 


36 


Limiting  Involute    Teeth. 


In  this  way  it  was  found  that  a  gear  of  four 
teeth  will  not  work  with  a  rack,  but  will  work 
with  an  internal  gear  having  a  number  of 
teeth  not  easily  calculated  with  existing  loga- 
rithmic tables,  but  which  is  approximately 
ten  thousand.  Also  that  a  pinion  of  three  j 
teeth  will  not  work  with  an  internal  gear 
having  more  than  3.56  teeth. 

For  unequal  teeth  we  can  use  the  formulae, 


2  TT  n 


tan.  7i  = 


tan.  H  = 


in  which  N  and  n  are  the  numbers  of  teeth 
in  the  pair  of  pointed  gears.  By  these  form- 
following  results  were  determined, 

N  K 

oo  0 


n 

2.695 
3. 

4. 
4.62 


10.17 
5.57 
4*62 


25°  27' 
33°  17 
34°  11 


71. — MINIMUM  NUMBERS  FOR  UNSYMMETRICAL   TEETH. 


If  we  drop  the  condition  that  the  fronts 
and  backs  of  the  teeth  shall  be  alike  we  have 
an  unimportant  case  that  is  similar  to  that 
already  studied,  but  much  more  intricate. 

If  we  carry  this  case  to  its  extreme,  and 
adopt  single  acting  teeth,  we  have  no  mini- 
mum numbers  at  all,  for  any  two  numbers 
of  teeth  will  then  work  together.  Fig.  52 
shows  one  tooth  working  with  three  teeth, 
and  any  other  combination  can  be  obtained. 
The  minimum  obliquity  for  a  given  pair  is 
obtained,  as  in  (66),  by  laying  off  the  known 
pitch  arc,  G  D,  at  right  angles  to  G  c,  and 
drawing  the  line  of  action  at  right  angles 
to  the  line  D  c.  The  obliquity  is  also  given 
by  the  formula : 

„  2    7T 

tan.  K  =  -=-. , 

N+n  ' 

in  which  n  and  N  are  the  numbers  of  teeth. 
When  the  obliquity  is  as  great  as  is  often 


Fig.  52. 


Unsymrnetrical 
teeth 


the  case  for  very  low  numbers  of  teeth  the 
action  may  be  impracticable  on  account  of 
the  great  friction  of  approach  (48).  The 
gears  of  Fig.  52  will  not  drive  each  other  on 
the  approach,  unless  the  tooth  surfaces  are 
very  smooth,and  the  power  transmitted  is 
almost  nothing. 


72. — MINIMUM  NUMBERS  FOR  GIVEN  ARC  OF  RECESS. 


It  has  generally  been  assumed,  although 
no  good  reason  for  the  assumption  has  ever 
been  given,  that  the  minimum  numbers  of 
teeth  occur  when  the  tooth  of  one  of  the 
gears,  Fig.  53,  is  pointed  at  the  interference 
point  /,  and  at  the  same  time  has  passed 
over  an  arc  of  recess  0  a  that  is  a  given  part 
of  the  whole  pitch  arc  a'  a. 

The  solution  is  simple  enough,  graphically 
by  repeated  trials,  or  by  a  formula  that  can 
be  applied  directly  without  the  usual  process 
by  trial  and  error. 

But,  as  involute  teeth  have  a  uniform  ob- 
liquity, there  is  no  necessity  for  assuming 


Fig. 

definite  arc  of  recess,  and  the  condition  on 


Involute  Efficiency. 


37 


which  the  problem  is  based  is  unwarranted.  I  spur  gears,  in  either  external  or  internal  con- 
No  real  limit  is  reached,  and  the  matter  is  I  tact,  in  the  Journal  of  the  Franklin  Institute 
not  worth  examination  at  any  length.  The  for  Feb.,  1888,  and  it  has  received  more  atten- 
problem  is  investigated,  for  both  bevel  and  I  tion  than  its  slight  importance  entitles  it  to. 


. — EFFICIENCY   OF   INVOLUTE   TEETH. 


But  little  can  be  said  in  addition  to  the 
matter  in  (49),  for  both  forms  of  teeth  in 
common  use  are  substantially  equal  with  re- 
spect to  the  transmission  of  power. 

From  the  formula  of  (49),  which  is  the 
formula  for  the  involute  tooth,  it  is  seen  that 
the  loss  from  friction  is  entirely  independent 
of  the  obliquity,  and,  therefore,  all  systems  of 
involute  teeth  are  independent  of  the  ob- 
liquity in  this  respect.  This  is  contrary  to  j 


the  accepted  idea  that  a  great  efficiency  re- 
quires a  small  obliquity. 

It  has  been  stated  on  high  authority  that 
the  involute  tooth  is  inferior  to  the  cycloidal 
tooth  in  efficiency,  but  the  statement  is  not 
true.  The  difference  in  efficiency  is  minute, 
a  small  fraction  of  one  per  centum,  but  what 
|  little  difference  there  is  is  always  in  favor  of 
the  involute  tooth. 


74. — OBLIQUITY   AND   PRESSURE. 


The  involute  tooth  action  is  in  the  direction 
of  the  line  of  action,  and  the  obliquity  is 
a  constant  angle.  It  is  variable  only  when 
the  shaft  center  distance  is  varied. 

As  the  pressure  is  always  equal  to  the 
product  of  the  tangential  force  at  the  pitch 
line  multiplied  by  the  secant  of  the  obliquity, 


(26),  it  is  constant  for  the  involute  tooth. 
Involute  teeth,  therefore,  have  a  steady  ac- 
tion that  is  not  possessed  by  other  forms; 
particularly  by  forms  which,  like  the  cy- 
cloidal, have  a  pressure  and  an  obliquity  that 
varies  between  great  extremes. 


75. — THE  ROLLER  OF  THE  INVOLUTE. 


The  involute  odontoid,  like  all  possible 
odontoids,  can  be  formed  by  a  tracing  point 
in  a  curve  that  is  rolled  on  the  pitch  line,  and 
this  roller  is  the  logarithmic  spiral  with  the 
tracing  point  at  its  pole,  (32). 

This  feature  is,  however,  more  curious  than 
useful,  and  it  is  not  of  the  slightest  im- 
portance in  the  study  of  the  curve.  Neither 
is  the  operation  of  rolling  the  involute  me- 
chanically possible,  for  the  logarithmic  roller 
has  an  infinite  number  of  convolutions  about 


its  pole,  and  the  tracing  point  would  never 
reach  the  pitch  line. 

The  involute  is  often  considered  to  be  a 
rolled  curve,  because  it  can  be  formed  by 
a  tracing  point  in  a  straight  line  that  rolls  on 
its  base  line;  but,  although  that  is  the  fact,  it 
is  a  special  feature  and  has  nothing  to  do 
with  the  rolled  curve  theory.  The  rolled 
curve  theory  requires  that  the  odontoid  shall 
be  forme  d  by  a  roller  that  rolls  on  the  pitch 
line  only . 


.    THE;    CYCLOIDAIv    SYSTEM. 


76. — THE   CYCLOIDAL   SYSTEM. 


If  the  curve  known  as  the  cycloid  is 
chosen  as  the  determining  rack  odontoid, 
(31),  the  resulting  tooth  system  will  be 
cycloidal. 

It  is  commonly  called  the  "  epicycloidal " 
system,  because  the  faces  of  its  teeth  are 
epicycloids,  but,  as  the  flanks  are  hypocy- 
cloids,  it  seems  as  if  the  name  "epihypo- 
cycloidal "  would  be  still  more  clumsy  and 
accurate. 

There  is  no  more  need  of  two  different 
kinds  of  tooth  curves  for  gears  of  the  same 


pitch  than  there  is  need  of  two  different 
kinds  of  threads  for  standard  screws,  or 
of  two  different  kinds  of  coins  of  the  same 
value,  and  the  cycloidal  tooth  would  never 
be  missed  if  it  was  dropped  altogether.  But 
it  was  first  in  the  field,  is  simple  in  theory,  is 
easily  drawn,  has  the  recommendation  of 
many  well-meaning  teachers,  and  holds  its 
position  by  means  of  "human  inertia,"  or 
the  natural  reluctance  of  the  average  human 
mind  to  adopt  a  change,  particularly  a 
change  for  the  better. 


77. — THE   CYCLOIBAL    TOOTH. 


The  cycloid  is  the  curve  A  that 
is  traced  by  the  point  p  in  the  circle 
C  that  is  rolled  on  the  straight  pitch 
line  p  I,  Fig.  54.  The  normal  at 
the  point  p  is  the  line  p  q  to  the 
point  of  tangency  of  the  rolling 
circle  and  the  pitch  line. 

The  line  of  action  is  the  circle  I  a, 
of  the  same  size  as  the  roller  C. 

As  no  tangent  arc  can  be  drawn  to 
the  line  of  action  from  the  pitch 
point  0  as  a  center,  no  terminal 
point  (18)  exists.  As  there  is  no 
point  upon  the  line  of  centers  from 
which  a  circle  can  be  drawn  tangent 
to  the  line  of  action,  there  will  be  no 
cusps,  (16)  except  on  the  pitch  line. 

The  cycloidal  tooth  can  be  drawn 
by  the  general  method  of  (24),  but 
there  are  several  easier  methods 
which  will  be  described.  There 
are  numerous  empirical  rules  and 
short  cuts  to  save  labor  and  spoil 
the  tooth,  which  will  not  be  de- 
scribed. 

When  the  pitch  line  is  of  twice  the  diame- 
ter of  the  line  of  action,  the  flank  of  the 
tooth  is  a  straight  line.  If  the  pitch  line  is 
less  than  twice  as  large  as  the  line  of  action, 
the  flank  of  the  tooth  will  be  under-curved, 


as  shown  by  Fig.  55,  and  it  is  customary  to 
avoid  the  resulting  weak  tooth  by  limiting 
the  line  of  action  to  a  diameter  not  greatei 
than  half  that  of  the  smallest  gear  to  be 
used. 


Cycloid al  Secondary  Action. 


78.— SECONDARY  ACTION. 


The  secondary  line  of  action  (21)  is 
a  circle,  Fig.  56,  differing  from  the 
pitch  circle  by  the  diameter  of  the 
primary  line  of  action,  either  inside  or 
outside  of  it. 

When  the  internal  secondary  line  of 
action  of  an  internal  pitch  line  coin- 
cides with  the  external  secondary  line 
of  action  of  its  pinion  _  there  will  be 
secondary  contact  between  the  gears, 
the  face  of  the  gear  working  with  the 
face  of  the  pinion  at  a  point  of  contact 
upon  the  combined  secondaries.  Fig. 
57  shows  this  for  the  cycloidal  tooth, 
the  two  faces  working  together  at  the 
point  a.  As  both  secondaries  are  cir- 
cles they  must  coincide,  and  the  sec- 
ondary action  will  be  continuous. 

When  the  teeth  are  also  in  contact  at 
b  on  the  primary  line  OL  action,  there 
will  be  double  contact. 


Undcrcurved  flanks 

Fig.  55. 


Secondary 


Fig  56. 


40 


Cycloidal  Interference. 


79. — INTERNAL    INTERFERENCE. 


If  the  secondary  lines  of  action  do  not 
come  together  the  teeth  will  not  touch  each 
other  at  all,  but  if  that  of  the  gear  is  smaller 
than  that  of  the  pinion  the  teeth  will  cross 
each  other  and  interfere.  The  line  c.  Fig. 
57,  is  the  face  of  the  gear  tooth,  and  the  line 
d  is  the  face  of  the  pinion  tooth  having  a 
primary  line  of  action  equal  to  the  difference 
between  the  pitch  lines.  The  secondary  line 
of  each  gear  coincides  with  the  pitch  line  of 
the  other,  and  the  faces  interfere  with  each 
other  the  amount  shown  by  the  shaded 
space. 

The  only  remedy  for  internal  interference 
is  to  reduce  the  diameter  of  the  primary  line 
of  action  to  half  the  difference  between  the 
diameters  of  the  pitch  lines,  or  else  to  leave 
off  one  of  the  faces  of  the  teeth. 

The  discovery  of  the  law  of  internal  cycloid- 
al  interference  is  due  to  A.  K.  Mansfield, 
who  published  it  in  the  ' '  Journal  of  the 
Franklin  Institute"  for  January,  1877.  It 
was  afterwards  re-discovered  by  Professor 
MacCord,  and  most  thoroughly  applied  and 
illustrated  in  his  "  Kinematics." 

When  interference  is  avoided  by  omitting 
one  of  the  faces  of  the  teeth  the  primary  line 
of  action  may  be  enlarged,  but  it  must  not 
then  be  larger  than  the  difference  between 
the  pitch  diameters. 

Fig.  58  shows  on  the  right  the  action 
when  the  face  of  the  gear  is  omitted,  and  on 
the  left  the  action  when  the  face  of  the  pin- 
ion is  left  off.  The  teeth  will  just  clear  each 
other,  each  one  touching  the  other  at  a  single 
point  a  in  its  pitch  line. 

As  the  contact  at  a  is  not  a  point  of  practi- 
cal action,  care  must  be  taken  that  the  arc  of 
action  at  the  primary  line  of  action  is  as 
great  as  the  circular  pitch,  for  otherwise,  as 
in  the  figure,  the  gears  will  not  be  in  continu- 
ous primary  action. 

The  rule  for  internal  interference,  simply 
stated,  is  that  the  diameters  of  the  pitch  lines 
must  differ  by  the  sum  of  the  diameters  of 
the  lines  of  action  if  the  teeth  have  both 
faces  and  flanks,  and  by  the  diameter  of  the 
acting  line  of  action  if  the  face  of  either  gear 
is  omitted.  For  the  standard  interchangeable 
system  the  gears  must  differ  by  twelve  teeth 


Fig  58 


if  both  teeth  have  faces,  and  by  six  teeth  if 
one  face  is  omitted. 

Fig.  62  shows  the  secondary  contact  in  the 
case  of  a  standard  internal  gear  of  twenty- 
four  teeth  working  with  a  pinion  of  twelve 
teeth,  and  it  is  to  be  noticed  that  the  teeth 
nearly  coincide  between  the  two  points  of 
contact.  Where  there  is  secondary  contact 
the  teeth  practically  bear  on  a  considerable 
line  instead  of  at  a  point. 


Cycloidal   Odontograph. 


41 


80. — THE   STANDARD   TOOTH. 

The  standard  tooth  (42),  selected  for  the 


cycloidal  system,  is  by  common  consent  the 
one  having  a  line  of  action  of  half  the  diame- 
ter of  a  gear  of  twelve  teeth,  so  that  that 
gear  has  radial  flanks. 

The  standard  adopted  by  manufacturers  of 
cycloidal  gear  cutters  is  that  having  radial 
flanks  on  the  gear  of  fifteen  teeth,  but  it  is 
not  and  should  not  be  in  use  for  other  pur- 


poses. If  any  change  is  made,  it  should  be 
made  in  the  other  direction,  to  make  the  set 
take  in  gears  of  ten  teeth. 

It  must  be  borne  in  mind  that  the  standard 
adopted  does  not  limit  the  set  to  the  stated 
minimum  number  of  teeth,  but  that  it  sim- 
ply requires  that  smaller  gears  shall  have 
weak  under-curved  teeth. 


81. — THE    ROLLED   CURVE   METHOD. 


It  happens  in  this  case,  and  in  this  case 
only,  that  the  rolled  curve  method,  which 
theoretically  applies  to  all  odontoids,  can 
be  actually  put  into  practical  use,  for  the 
generating  roller  is  here  the  circle,  the  sim- 
plest possible  curve. 

As  in  Fig.  59,  roll  a  circle  of  the  .diameter 
of  the  circle  of  action  upon  the  outside  of  the 
pitch  line  for  the  faces,  and  upon  the  inside 
for  the  flanks,  and  a  fixed  point  in  it  will 
trace  the  curve. 

The  method  can  be  used  by  actually  con- 
structing pitch  and  rolling  circles,  but  the 
same  result  can  be  reached  more  easily  and 
quite  as  accurately  by  drawing  several  cir- 
cles, and  then  stepping  from  the  pitch  point 
along  the  pitch  line,  and  back  on  the  circles 
to  the  desired  point.  If  the  length  of  the 


Construction  by  rolling 

Fig.  59. 

step  is  not  more  than  one-tenth  of  the  diam- 
eter of  the  circle,  the  error  will  not  be  over 
one-thousandth  of  an  inch  for  each  step. 

This  method  is  the  best  one  to  adopt,  ex- 
cept for  the  standard  tooth. 


82. — THE   THREE  POINT   ODONTOGRAPH. 


It  is  a  simple  matter  to  draw  the  tooth 
curve  by  means  of  rolling  circles,  but  such 
a  method  requires  skill  on  the  part  of  the 
draftsman.  It  is,  moreover,  nothing  but  a 
method  for  finding  points  in  the  curve  for 
which  approximate  circular  arcs  are  then 
determined. 

The  "three  point"  odontograph  is  sim- 
ply a  record  of  the  positions  of  the  centers 
of  the  circles  which  approximate  the  most 
closely  to  the  whole  curve  of  the  standard 
tooth.  The  positions  of  two  .points,  a  at  the 
center  of  the  face  or  of  the  flank,  Fig.  60, 
and  b  at  the  addendum  point  or  root  point 
of  the  curve,  were  carefully  computed,  and 
then  the  position  of  the  center  C  of  the 
circle  which  passes  through  these  two 


points  and  the  pitch  point  0,  was  calcu- 
lated. The  circle  that  passes  through  these 
three  points  is  assumed  to  be  as  accurately 
approximate  to  the  true  curve  as  any  pos- 
sible circular  arc  can  be. 

The  odontograph  gives  the  radius  "  rad." 
of  the  circular  arc,  and  the  distance  "dis." 
of  the  circle  of  centers  from  the  pitch  line, 
for  the  tooth  of  a  given  pitch,  and  their 
values  for  other  pitches  are  easily  found  by 
simple  multiplication  or  division. 

The  advantages  of  this  method  lie  in  the 
facts  that  the  desired  radius  and  distance 
are  given  directly,  without  the  labor  of  find- 
ing them,  and  that  as  they  are  computed 
they  are  free  from  errors  of  manipulation. 
In  point  of  time  required,  the  advantage  is 


\ 


42 


Cycloidal   Odontograph. 


WeofJW* 


with  the  odontograph  in   the  ratio  of 
ten  to  one. 

The  greatest  error  of  the  odonto 
graphic  arc,  shown  greatly  exaggerated 
by  the  dotted  lines,  is  at  the  point  c  on 
the  face,  and  it  is  greater  on  a  twelve- 
toothed  pinion  than  on  any  larger  gear. 
For  a  twelve-toothed  pinion  of  three- 
inch  circular  pitch,  a  large  tooth,  the 
actual  amount  of  the  maximum  error  is 
less  than  one  one-hundredth  of  an  inch, 
and  its  average  for  eight  equidistant 
points  on  the  face  is  about  four-thousandths  I  that  stated  will  be  due  to  manipulation,  and 
of  an  inch.  Any  error  that  is  greater  than  !  not  to  the  method. 


To  apply  the  odontograph  to  any  particu 
lar  case,  tirst  draw  the  pitch,  addendum 


83. — USING   THE   ODONTOGRAPH. 

distance  "dis."  inside  of  it.    Take  the  face 
radius  "rad."on  the  dividers,  and  draw  in 


root,  and  clearance  lines,  and  space  the  pitch 
line,  Figs.  60  and  61. 

Then  draw  the  line  of  flank  centers  at 
the  tabular  distance  "dis."  outside  of  the 
pitch  line,  and  the  line  of  face  centers  at  the 


all  the  face  curves  from  centers  on  the  line 
of  face  centers;  then  take  the  flank  radius 
"rad."and  draw  all  the  flank  curves  from 
centers  on  the  line  of  flank  centers. 


THREE  POINT    ODONTOGRAPH. 
STANDARD  CYCLOIDAL  TEETH. 

INTERCHANGEABLE  SERIES. 
From  a  Pinion  pf  Ten  Teeth  to  a  Rack. 


For  One 

/***  For  One  Inch 

DIAMETRAL  PITCH. 

CIRCULAR  PITCH. 

NUMBER  OF 
TEETH 

For  any  other  pitch  divide  by 
that  pitch. 

For  any  other  pitch  multiply  by 
that  pitch. 

IN  THE  GEAR. 

Faces. 

Flanks. 

Faces. 

Flanks. 

Exact. 

Intervals. 

Rad. 

Dis. 

Rad. 

Dis. 

Rad. 

I  is. 

Rad. 

Dis. 

10 

10 

1.99 

.02 

—  8.00 

^ 

.62 

.01 

—2.55 

1.27 

11 

11 

2.00 

.04 

—  11.05 

6.50 

.63 

.01 

-3.34 

207 

12 

12                 201 

06 

oo 

'      00 

.64 

.02 

00 

oo  : 

13—14 

2.04 

.07 

15.10           9.43 

.65 

.02 

4.80 

3.00 

15Vi 

15—16 

2.10 

.09 

7.86 

'   3  46 

.67 

.03 

2.50 

1  10 

ITji 

17-18 

2.14 

.11 

6.13 

2.20 

.68 

.04 

1.95 

.70 

20 

19-21 

2.20 

.13 

5.12  / 

1.57 

.70 

.04 

.63 

.50 

23 

22-24 

2.26 

.15 

4.5QV 

1.13 

.72 

.05 

.43 

.36 

27 

25-29 

2.33 

.16 

4.10 

.96 

.74 

.05 

30 

.29 

83 

30-36 

2.40 

.19 

3.80 

.72 

.76 

.06 

.      .20 

.23 

42 

37-48 

2.48 

.22 

3.52 

.63 

.79 

.07 

.12 

.20 

58 

49—72 

2.60 

.25 

3.33 

.54 

.83 

.08 

.06 

,    .17 

97 

73—144 

2.83 

.28 

3.14 

.44 

.90 

.09 

1.00 

.14 

290 

145-300 

2.92 

.31 

3.00 

.38 

.93 

.10 

.95 

.12 

00 

Rack 

2.96 

.84 

2.96 

.34 

.94 

.11 

.94 

.11 

Cyclo  idal  Odo  n  tograph . 


The  table  gives  the  distances  and  radii  if 
the  pitch  is  either  exactly  one  diametral  or 
one  inch  circular,  and  for  any  other  pitch 
multiply  or  divide  as  directed  in  the  table. 

Fig.  61  shows  the  process  applied  to  a 
practical  case,  with  the  distances  given  in 
figures. 


Fig.  62  shows  the  c'ame  process  applied  to 
an  internal  gear  of  twenty-four  teeth  work- 
ing with  a  pinion  of  twelve  teeth.  It  illus- 
trates secondary  action  and  double  contact. 
It  also  shows  the  actual  divergence  of  the 
Willis  odontographic  arp  from  the  true 
curve. 


Odontographic  example 
Fig.  61. 


Internal  teeth 

Fig.  62. 


44 


Willis   Odontograph. 


84. — THE  WILLIS   ODONTOGRAPH. 


This  is  the  oldest  and  best  known 
of  all  the  odontographs,  but  it  is 
inferior  to  several  others  since  pro- 
posed, not  only  in  ease  of  operation, 
but  in  accuracy  of  result. 

To  apply  it,  find  the  pitch  point* 
a  and  a'  half  a  tooth  from  the  pitch 
point  0,  Fig.  63,  draw  the  radii  a  c 
and  a'  c',  lay  off  the  angles  cab  and 
c'  a'  b',  both  75°,  and  Jay  off  the 
distances  a  b  and  a'  b'  that  are  given 
by  table. 

The  centers  b  and  b'  thus  found  are 
the  centers  of  circular  arcs  that  are 
tangent  to  the  tooth  curves  at  d 
and  d'.  The  dividers  are  set  to  the 
radius  b  0  or  b'  0  to  draw  the  curves. 

The  Willis  arc  touches  the  true  curve  only 
at  the  pitch  point  0,  and  its  variation  else- 
where is  small,  but  noticeable.  On  the  face 
of  the  tooth  of  a  twelve-toothed  pinion  of 
three  inch  circular  pitch,  its  error  at  the  ad- 
dendum point  is  four-hundred  ths  of  an  inch, 
and  it  will  average  three  times  that  of  the 
three  point  method  (82).  The  error  is  shown 
by  Fig.  62. 

The  greatest  error  of  the  method  is  due  to 
manipulation.  The  angle  is  usually  laid  off 
by  a  card,  and  the  center  measured  in  by  a 
scale  on  the  card.  The  circle  of  centers  is 


TJie  Willis  odontograph 

Fig.  63. 


then  drawn  through  the  center,  and  unless 
great  care  is  used  the  chances  of  error  are 
great. 

180 
The  angle  90°— c  ab  =  JFs=— ,  and  the 


-  sin.  W,  in  which 


distance  a  b  =  5— • - 

27T    t 

is  the  number  of  teeth  in  the  gear  of  the 
same  set  which  has  radial  flanks,  usually 
12  ;  c  is  the  circular  pitch,  and  t  is  the  num- 
ber of  teeth  in  the  gear  being  drawn.  The 
positive  sign  is  used  for  the  face  radius,  and 
the  negative  for  the  flank  radius. 


85.  —  KLEIN'S  CO-ORDINATE  ODONTOGRAPH. 

This  is  a  method  of  finding  the  positions 
of  several  points  on  the  tooth  curve  by 
means  of  their  co-ordinates  referred  to  axes 
through  the  pitch  point.  Any  point  on  the 
curve  is  found  by  laying  off  a  certain  dis- 
tance on  the  radius  Y,  Fig.  64,  and  then 
a  certain  distance  at  right  angles  to  it,  the 
distances  being  given  by  a  table  for  a  certain 
standard  tooth. 

As  many  points  as  required  are  found  by 
this  method,  and  then  the  curve  is  drawn  in 
by  curved  rulers,  or  by  finding  the  approxi- 
mating circular  arc. 

This  odontograph  is  to  be  found  in  Klein's 
Elements  of  Machine  Design. 


Coordinate  odontograpli 

Fig.  64. 


Obliquity  of  Action. 


86. — THE  TEMPLET  ODONTOGRAPH. 


Prof.  Robinson's  templet  odontograph  is 
an  instrument,  not  a  method.  It  is  a  piece 
of  sheet  metal,  Fig.  65,  having  two  edges 
shaped  to  logarithmic  spirals.  It  is  laid 
upon  the  drawing,  according  to  directions 
given  in  an  accompanying  pamphlet,  and 
used  as  a  ruler  to  guide  the  pen.  It  can  be 
fastened  to  a  radius  bar,  and  swung  on  the 
center  of  the  gear,  to  draw  all  the  teeth. 
See  Van  Nostrand's  Science  Series,  No.  24, 
for  the  theory  of  the  instrument  in  detail. 


The  templet  odontograph 

Fig.  65. 


87. — OBLIQUITY  OF   THE  ACTION. 


When  the  point  of  contact  between  two 
teeth  is  at  the  pitch  point  0,  Fig.  66,  the 
pressure  between  the  teeth  is  at  right  angles 
to  the  line  of  centers,  but,  as  the  point  of  con- 
tact recedes  from  the  line,  the  direction  of 
the  pressure  varies  by  an  angle  of  obliquity 
which  increases  from  zero  until  the  point  K, 
at  the  intersection  of  the  addendum  circle 
with  the  line  of  action,  is  reached. 

The  angle  K  =  K  0  W,  of  the  maximum 
obliquity,  can  be  found  by  solving  the  trian- 
gle C  c  K,  and  for  the  standard  set  we  have, 

cos.  2  K  =  ^      ^, 
3  n-\-  18 

in  which  n  is  the  number  of  teeth  in  the 
gear. 

For  the  smallest  gear  of  the  set,  the  one 
having  twelve  teeth,  K  is  20°  15',  and  for  the 
rack  it  is  24°  5',  so  that  it  will  always  be  be- 
tween those  two  limits  for  external  gears, 
and  greater  for  internal  gears. 

The  friction  between  two  gear  teeth  in- 
creases with  the  angle  of  obliquity,  but  not 


w 


Obliquity 

Fig.  66. 

in  direct  proportion.  With  the  involute 
tooth  the  work  done  while  going  over  a  cer- 
tain arc  from  the  line  of  centers  is  propor- 
tional to  the  square  of  the  arc,  and  for 
cycloidal  teeth  the  increase  with  the  arc  is 
still  more  rapid.  Therefore  it  is  the  maxi- 
mum obliquity  of  the  action  that  principally 
determines  the  injurious  effects  of  friction. 


— THE   CUTTER    LIMIT. 


When  the  number  of  teeth  in  the  gear  is 
less  than  that  in  the  gear  having  teeth  with 
radial  flanks,  the  flanks  will  be  under-curved, 
and  when  too  much  so  they  cannot  be  cut 
with  a  rotary  cutter.  The  teeth  of  Fig.  55 
could  not  be  cut  with  a  rotary  cutter  beyond 
the  points  where  the  tangents  to  the  two 
sides  are  parallel. 

The  limit  is  reached  when  the  last  point 


that  is  cut  by  the  rotary  cutter  is  also  the 
last  point  that  is  touched  by  the  tooth  of  the 
rack  in  action  with  it,  not  allowing  for  in- 
ternal gears. 

The  diameter  of  the  gear  when  this  limit 
is  reached  is  found  by  the  formula, 

-»"»  rt     T  C 


46 


Limiting   Cycloidal     Teeth. 


in  which  D  is  the  diameter  of  the  gear,  d  is 
the  diameter  of  the  circle  of  action,  c  is  the 
circular  pitch,  and  a  is  the  addendum 

For  the  common  addendum  of  unity 
divided  by  the  diametral  pitch  this  may  be 
put  in  the  shape,  ^ 

n  =  s i 


in  which  *  is  the  number  of  teeth  in  the 
radial  flanked  gear,  and  n  is  the  number  in 
the  required  cutter  limit. 

For  the  common  series,  where  8  =  12,  we 
have  n  =  8.26;  and  for  the  cutter  standard  of 
*  =  15,  we  have  n  =  10.80,  so  that  cutters 
could  easily  be  made  to  cut  gears  with  less 
than  s  teeth. 


When  the  rolling  circle  for 
the  faces  is  of  half  the  diam- 
eter of  the  pitch  line  of  the 
mating  gear,  the  flanks  of 
both  gears  will  be  straight 
radial  lines,  as  in  Fig.  67. 

Such  gears  are  fitted  to  each 
other  in  pairs,  and  are  not 
interchangeable  with  other 
sizes.  Their  teeth  are  more 
easily  made  than  those  of 
standard  gears.  The  maxi- 
mum obliquity  is  less,  but 
the  strength  of  the  teeth  is 
also  less  than  usual.  There 
is  no  reason  for  making  such 
teeth  in  preference  to  the 


).— RADIAL  FLANKED  TEETH. 


standard,    al- 


though, for  that  reason  probably,  they  are 
used  to  a  considerable  extent.    It  would  be 


Radial  flanks 

Fig.  67. 


difficult  to  devise  a  form  of  tooth  so  whimsi- 
cal that  it  would  find  no  one  to  adopt  and 
use  it.  J  £ 


90. — THE  LIMITING  NUMBERS   OP  TEETH. 


When  the  number  of  teeth  in  a  driving 
gear  is  small,  the  point  p,  Fig.  68,  of  its 
pointed  tooth  may  go  out  of  action  by  leav- 
ing the  line  of  action  0  g  before  a  certain 
definite  arc  of  recess  0  r  has  been  passed  over, 
and  the  problem  is  to  find  the  smallest  num- 
ber of  teeth  in  the  following  gear  that  will 
just  allow  the  given  recess. 

This  question,  which  is  not  a  particularly 
important  one,  is  discussed  at  length,  and 
applied  to  both  bevel  and  spur  gears,  in  either 
external  or  internal  contact,  in  an  article  in 
the  "Journal  of  the  Franklin  Institute"  for 
Feb.,  1888,  and  we  will  here  consider  only 
the  case  of  the  common  spur  gear. 

The  recess  0  r  is  given  as  a  times  the  cir- 
cular pitch,  and  the  thickness  a  r  of  the 
tooth  is  given  as  b  times  the  same.  The 
diameter  of  the  circle  of  action  is  q  times 


Limiting  ttftfth 

Fig.  68. 


that  of  the  pitch  line  of  the  following  gear. 
The  number  of  teeth  in  the  driving  gear  is  d, 
and  the  number  in  the  following  gear  is  /. 


The  Pin    Tooth. 


47 


M  is  an  auxiliary  angle  equal  to  —     -,    and 
W  is  an  angle  —  /«  —  —  \. 

Then  the  required  number  /  can  be  found 
by  a  process  of  trial  and  error  with  the 
formula, 


sin.  (M  -f  W) 
sin.  W 


-Tf-M 


For  an  example,  let  the  recess  be  £  of  the 
pitch,  the  tooth  equal  to  the  space,  and  the 
flanks  of  the  follower  to  be  radial.  Let  the 
problem  be  to  find  a  follower  for  a  driver  of 
seven  teeth.  This  gives  a  =  £ ,  b  =  $,' q  =  i, 
d  =  7,  and  the  formula  becomes 


gin. 


in.  l^L  +  25°  43'  \ 
\  J  i 


sin.  25°  43' 


If  we  put  /  at  random,  at  20,  we  shall 
get,  +.134  =  0.  Next,  trying/ =10,  we 
get,  —  .132  =  0,  and  the  opposite  signs  show 
that  /  is  between  20  and  10.  Trying  12  the 
result  is  positive,  and  for  11  it  is  negative, 
showing  that  12  is  the  required  value  of  /. 
That  is,  7  teeth  will  not  drive  less  than  12 
teeth  with  radial  flanks,  unless  it  is  allowed 
an  arc  of  recess  greater  than  f  of  the  pitch. 

For  another  example,  test  MacCord's  value 
of  382  as  the  least  driver  for  a  follower  of 
10  teeth,  when  recess  equals  the  pitch  and 
the  follower  has  radial  flanks.  Trying  d  — 
382,  the  error  is  negative  ;  for  383  it  is  also 
negative,  but  for  384  it  is  positive,  and  there- 
fore the  latter  is  the  true  number. 

Extensive  and  sufficiently  accurate  tables  of 
limiting  values  are  given  by  MacCord  in  his 
"Kinematics." 


5. 


1PITST    TOOTH     SYSTEM. 


91. — THE  PIN  GEAR  TOOTH. 


The  theory  of  the  pin  gear  tooth  is  en- 
tirely beyond  the  reach  of  the  "  rolled  curve" 
method  of  treatment,  and,  therefore,  writers 
who  have  adopted  that  method  have  had  to 
depend  more  on  special  methods  adapted  to 
it  alone  than  on  general  principles.  The  re- 
sult is  that  its  properties  are  often  given  in- 
correctly, or  with  an  obscurity  and  complica- 
tion that  is  bewildering  to  the  student. 
Although  the  tooth  is  one  of  the  oldest  in 
use,  its  theory  is  so  difficult  that  its  defect 
was  not  discovered  until  within  a  very  few 
years,  by  MacCord,  about  1880,  and  it  was 


not  until  it  was  examined  by  means  of  its 
normals  that  a  remedy  for  that  defect  was 
discovered. 

By  treating  the  curve  on  the  general  prin- 
ciples here  adopted,  as  a  special  form  of  the 
segmental  tooth,  it  can  be  studied  with  ease, 
and  its  peculiarities  developed  in  a  complete 
and  satisfactory  manner.  The  method,  in 
general  terms,  is  to  find  the  conjugate  tooth 
curve  of  the  gear,  for  the  given  circular  tooth 
curve  of  the  pinion,  and  it  presents  no  new 
features  or  difficulties. 


92. — APPROXIMATE  FORM   OF  PIN   TOOTH   CURVE. 


Considered  roughly,  but  accurate  enough 
for  teeth  of  small  size,  the  form  of  the 
gear  tooth  b,  Fig.  69,  is  a  simple  parallel 
to  the  epicycloid  E,  formed  by  the  center  e 
of  the  pin,  and  is  to  be  drawn  tangent  to 


any  convenient  number    of    circles  having 
centers  on  the  epicycloid. 

The  action  is  practically  all  on  one  side  of 
the  line  of  centers,  the  face  of  the  gear  tooth 
working  with  the  part  of  the  pin  that  is 


The  Pin    Tooth. 


inside  of  its  pitch  line.  It  is,  therefore,  all 
approaching  action  when  the  pin  drives  and 
all  receding  action  when  the  gear  drives,  and 
it  is  best  to  avoid  the  increased  friction  of 
the  approaching  action  by  always  putting  the 
pins  on  the  follower. 


Zantern  wheel 

Fig.  70. 


Pin  gearing 

Fig.  69. 


93. — ROLLER  TEETH. 


The  pin  gear  is  particularly  valuable  when 
the  pins  can  be  made  in  the  form  of  rollers, 
Fig.  70,  for  then  the  minimum  of  friction  is 
reached.  The  roller  runs  freely  on  a  fixed 
stud,  or  on  bearings  at  each  end,  and  can  be 
easily  lubricated. 

The  friction  between  the  tooth  and  pin, 
otherwise  a  sliding  friction  at  a  line  bearing, 
is,  with  the  roller  pin,  a  slight  rolling  fric- 
tion, and  the  sliding  friction  is  confined  to 


the  surface  between  the  roller  and  its  bear- 
ings. 

When  the  roller  pin  is  used  there  can  be 
no  increased  friction  of  approach,  and  the 
pin  wheel  can  drive  as  well  as  follow. 

For  very  light  machinery,  such  as  clock 
work,  there  is  no  form  of  tooth  that  is  su- 
perior to  the  roller  pin  tooth,  and,  with  the 
improvement  to  be  explained,  there  is  no 
better  form  for  any  purpose. 


94. — CUTTING  THE  PIN  TOOTH. 

The  pin  gear  tooth  can  be  very  easily  and 
accurately  shaped  by  mounting  a  revolving 
milling  cutter  M,  Fig.  71,  of  the  size  of  the 
pin,  upon  a  wheel  A,  and  causing  it  to  roll 
with  a  wheel  B,  carrying  the  gear  blank  Q. 
The  mill  will  shape  the  teeth  to  the  correct 
form. 


Pin  gear  cutter 

Fig.  71. 


95. — PARTICULAR  FORMS  OF  PIN  GEARS. 


When  the  pins  are  supported  between 
two  plates,  as  in  Fig.  70,  the  wheel  is  called 
a  "lantern"  wheel,  and  is  the  most  common 


form  of  clock  pinion.  The  pins  are  some- 
times called  "staves,"  and  are  sometimes 
known  as  "leaves." 


Defect  of  Pin    Tooth. 


4<J 


When  the  diameter  of  the  pin  is 
zero,  Fig.  72,  it  being  merely  a 
point,  the  correct  tooth  curve  will 
be  a  simple  epicycloid. 

When  the  pin  gear  is  a  rack,  Fig. 
73,  the  tooth  bears  on  the  pin  only 
at  a  single  point  on  the  pitch  line, 
and  the  action  is  therefore  very  de- 
fective unless  the  roller  form  of  pin 
is  used.  This  form  is  more  properly 
a  particular  case  of  the  Involute  tooth, 
for  the  shape  of  the  pin  is  immaterial  if  it  does 
not  interfere  with  the  gear  tooth.  The  circle 
with  center  on  a  straight  line  is  not  an 
odontoid  at  all,  for,  although  it  coincides  as 
a  whole  and  for  a  single  instant  with  a  cir- 
cular space  in  the  gear,  it  has  no  proper  and 
continuous  tooth  action. 

The  gears  of  Fig.  74,  sometimes  classed 
with  pin  gearing,  are  not  pin  gears  at  all. 
An  epicycloidal  face  working  with  a  radial 
flank  is  a  very  common  combination. 

When  the  diameter  of  the  pin  wheel  is  half 
that  of  the  internal  gear  with  which  it  works, 
we  have  the  combination  of  Fig.  75.  The 
pins  may  run  in  blocks  fitted  to  the  straight 
slots. 


Point  gears 

Fig.  72. 


fin  rack 

Fig.  73. 


Rot  pin  gears 

Fig.  74. 


Radial  pin  teeth 

Fig.  75. 


). — CORRECT  FORM   AND   DEFECT   OF   PIN   TEETH. 


Although  the  pin  tooth  is  apparently  of  a 
very  simple  form,  a  close  examination  will 
show  that  it  is  really  quite  complicated,  and 
that  its  practical  action  is  incomplete  and  de- 
fective. There  is  a  cusp  (16),  and  conse- 
quent failure  in  the  action,  that  is  of  small 
importance  when  the  teeth  are  small,  but 
which  is  troublesome  when  they  are  large. 
This  defect  need  not  be  considered  when 
pinions  for  clock  work  are  in  view,  but  if 
pin  wheels  are  to  be  used  for  large  machinery 
and  heavy  power  it  is  important. 

If  the  pin  a,  Fig.  76,  is  examined  as  an 
odontoid,  it  will  be  seen  lhat  it  is  a  true 
odontoid  only  within  the  line  TeT  that  is 
tangent  to  the  pitch  line  at  the  center  of  the 
pin,  for  all  normals,  as  pe,  from  points  out- 
side of  that  line,  intersect  the  pitch  line  at 
the  center. 

Drawing  the  normals,  which  are  radii  of 
the  pin,  we  can  easily  construct  the  line  of 


action  and  the  conjugate  tooth  curve.  The 
line  of  action,  commencing  at  the  pitch  point 
0,  Fig.  77,  is  there  tangent  to  the  line  eOm, 
which  passes  through  the  center  e  of  the  pin, 
curves  toward  Oh,  the  tangent  to  the  pitch 
line  at  the  pitch  point,  and  touches  it  at  the 
point  h,  at  the  distance  Oh,  equal  to  the  ra- 
dius of  the  pin.  From  the  point  h  it  follows 
the  circle  hJh'  to  the  point  h' ,  thence  return- 
ing to  the  pitch  point  and  forming  the  loop 
OKL. 

From  the  center  c  of  the  gear,  Fig.  78,  we 
can  always  draw  a  tangent  arc  FN  to  the 
line  of  action  at  the  point  F,  and  therefore 
there  will  always  be  a  cusp  at  .ZV  on  the 
tooth  curve.  The  tooth  curve  must  end  at 
the  cusp,  and,  to  avoid  interference,  the  pin 
must  be  cut  off  at  the  arc  W,  drawn  through 
the  point  F,  from  the  center  C. 

The  whole  pin  is  generally  used,  and 
when  it  is  a  roller  it  must  be  whole,  and 


Improved  Pin    Tooth, 


then  interference  can  be  avoided  only  by 
cutting  away  the  tooth  curve  until  it  will  al- 
low it  to  pass. 

The  complete  tooth  curve  has  a  first 
branch  NOM,  Fig.  78,  which  is  the  only 
part  that  can  be  used,  an  in  operative  second 
branch  from  the  first  cusp  N  to  the  second 
cusp  Q  on  the  arc  EQ,  and  thence  an  inopera- 
tive circle  ORQ'. 


JLine  of  action 

Fig.  77. 


'Correct  action 

Fig.  78. 


97. — AN   IMPROVED   PIN   TOOTH. 


The  cause  of  the  broken  action  of  the  pin 
tooth  is  the  cusp,  which  is  always  present 
when  the  center  of  the  pin  is  on  the  pitch 
line,  and  it  can  be  avoided  by  placing  the 
center  back,  as  in  Fig.  79,  to  such  a  distance 
inside  the  pitch  line  that  the  cusp  does  not 
occur. 

When  the  center  of  the  pin  is  inside  the 
pilch  line,  the  whole  circle  of  the  pin  is  a  true 
odontoid,  and  the  distance  en  of  the  center 
from  the  pitch  line  can  be  so  chosen  that 
the  cusp  is  not  formed. 

This  distance  does  not  appear  to  be  sub- 
ject to  any  simply  stated  rule,  but  in  the 
single  case  of  the  pin  rack  it  is  determined  by 
the  formula: 

2      d* 
27    D   ' 


x  — 


in  which  x  is  the  required  distance  en,  D  is 
the  diameter  of  the  gear,  and  d  is  the  diame- 
ter of  the  pin. 


If  the  angle  CeO,  Fig.  79,  is  not  less  than 
a  right  angle,  there  will  be  no  cusp  on  the 


Fig.  ?9. 


90 

Corrected  pin  year 


gear  tooth  if  the  diameter  of  the  gear  is 
greater  than  that  of  the  pin. 


6.    TWISTED,     SPIRAL,,    AND    \VORN1 


5.— STEPPED   GEARS. 


When  two  or  more  gears,  Fig.  80,  of  the 
same  pitch  diameter,  are  placed  in  contact  en 
the  same  shaft,  they  will  evidently  act  as  in- 
dependently of  each  other  as  if  they  were 
some  distance  apart,  while  they  appear  to  act 
together  as  a  single  gear  with  irregular  teeth. 
They  are  known  as  "  Hooke's  Gears." 

It  matters  not  how  many  different  kinds  or 
numbers  of  teeth  the  several  gears  may  have, 
or  in  what  order  they  are  arranged,  if  those 
that  work  together  on  opposite  shafts  are 
matched.  They  may  be  given  an  irregular 
arrangement,  as  in  Fig.  80  ;  a  spiral  arrange- 
ment, as  in  Fig.  81 ;  a  double  spiral,  or  "her- 
ring-bone "  arrangement,  as  in  Fig.  82  ;  a  cir- 
cular arrangement,  as  in  Fig.  83,  or  other- 
wise at  will. 


Stepped  f/ear 


Spiral  arrangement 

Fig.  81. 


Double  spiral  arrangement 

Fig.  82. 


Circular  arrangement 

Fig.  83. 


99.— TWISTED   TEETH. 


The  thickness  of  the  component  gears  has 
nothing  to  do  with  the  theoretical  action  of 
the  stepped  gear  as  a  whole,  and  therefore 
we  can  have  them  as  thin  as  required.  If  the 
thickness  is  infinitesimal  the  component 
character  of  the  gear  is  not  apparent,  and  it 
is  known  as  a  twisted  gear,  Fig.  84. 

When  the  teeth  are  twisted  there  -may  al- 
ways be  one  or  more  points  of  contact  at  the 
line  of  centers,  where  the  theoretical  fric- 
tion is  nothing,  and  therefore  they  are  par- 
ticularly well  suited  for  rough  cast  teeth. 
Furthermore,  if  the  teeth  are  badly  shaped 


Tivisted  arrangement 

Fig.  84. 


52 


Twisted   Teeth. 


the  twisted  arrangement  tends  to  distribute 
the  errors  so  that  they  are  not  as  noticeable. 
The  oblique  action  of  twisted  teeth  tends 
to  produce  a  longitudinal  motion  of  the 
gears  upon  their  shafts,  which  must  be 
guarded  against.  This  end  thrust  may  be  j 
avoided  by  so  forming  the  twist  that  there  ! 


are  aiways  two  oblique  bearings  between  the 
teeth,  acting  in  opposite  directions,  as  in  the 
herring-bone  arrangement. 

The  twisted  form  of  tooth  is  seldom  found 
in  practice,  except  in  the  form  of  spiral  and 
double  spiral  teeth,  for  the  difficulty  of  form- 
ing other  twists  is  great. 


100. — EDGE   TEETH. 


If  the  twist  of  the  twisted  tooth  is  such 
that  some  part  of  the  twist  at  the  pitch  cylin- 
der is  always  upon  the  line  of  centers,  the 
gears  will  always  be  in  action  whether 
there  are  full  teeth  or  not,  and  they  will 
work  with  theoretical  accuracy  if  they  are 
reduced  to  edges  in  the  pitch  cylinder,  as  in 
Fig.  85. 

The  friction  of  the  edge  tooth  is  theoret- 
ically nothing,  as  there  is  no  sliding  of  the 
teeth  on  each  other.  There  is  but  one  point 
of  contact,  and  that  is  always  upon  the  line 
of  centers;  but  if  any  power  is  carried  the 
pressure  will  soon  destroy  the  single  point  of 
contact. 


Fig.  85. 


If  the  edges  are  thick  the  action  will  be 
stronger,  but  there  will  still  be  but  one  point 
of  contact. 


101. — INVOLUTE   TWISTED  TEETH. 

When  the  form  of  the  tooth  is  the  invo- 1  point  contact.     The  straight  involute  tooth 


lute,  and  the  twist  is  such  that  some  part  of 
it  on  the  pitch  cylinder  always  crosses  the 
line  of  centers,  the  teeth  will  remain  in  con- 
tact, when  the  parallel  axes  are  separated, 
until  their  points  are  separated,  although  the 
contact  may  sometimes  be  very  short  or  even 


will  fail  as  soon  as  the  arc  of  contact  is  less 
than  the  tooth  arc. 

Twisted  involute  teeth  are  therefore  partic- 
ularly valuable  for  gears  for  driving  rolls,  or 
for  other  purposes  where  the  shaft  distance 
is  variable. 


102. — FORMATION    OF    THE   TWISTED   TOOTH. 


When  the  twist  is  a  uniform  spiral  there 
are  convenient  methods  for  shaping  the 
tooth,  but  the  twisted  tooth  in  general  can  be 
formed  only  by  the  processes  of  (27),  (28)  and 
(29),  and  then  only  when  the  twist  is  not 
very  irregular. 

The  principle  of  the  linear  planing  opera- 


tion of  (29)  is  the  same  as  for  the  straight 
tooth,  but  the  blank  must  be  rotated  accord- 
ing to  the  form  of  the  twist  adopted,  while 
the  tool  is  cutting.  The  twisting  motions 
are  independent  of  the  feeding  motion,  and 
are  repeated  at  every  stroke. 


103. — SPIRAL   GEARS. 


The  spiral  gear  is  that  particular  form  of 
the  twisted  gear  which  has  uniformly 
twisted  teeth,  and  it  is,  therefore,  a  particu- 


lar form  of  the  common  spur  gear.  It  has 
such  peculiar  properties  that  it  is  often 
classed  by  itself  as  a  separate  form  of  tooth. 


Spiral  Act! 072. 


The  normal  spiral  section  is  that  section  of 
the  teeth  of  the  spiral  gear  that  is  made  by  a 
spiral  surface,  called  a  helix,  that  is  at  right 
angles  with  the  teeth.  It  is  the  equivalent, 
for  spiral  teeth,  of  the  normal  section  of  the 
spur  gear  that  is  made  by  a  plane,  or  of  the 
normal  section  of  the  bevel  gear  that  is  made 
by  a  sphere.  As  with  spur  and  bevel  gears, 
the  action  of  the  teeth  on>  each  other  should 
be  studied  upon  this  normal  surface.  As  the 
helix  cannot  be  ^represented  upon  a  plane 
figure  it  must  be  imagined,  and  as  it  is  ob- 
scure it  requires  close  attention. 

Any  two  spiral  teeth  will  work  together, 
provided  their  normal  spiral  sections  are  con- 
jugate (24),  and,  as  the  shape  of  the  normal 
spiral  section  is  independent  of  the  angle  of 


the  spiral,  two  spiral  gears  will  work  to- 
gether, approximately,  on  shafts  that  are 
askew.  This  will  be  seen  more  clearly  if  the 
spiral  section  is  imagined  to  be  a  flexible 
sheet-metal  toothed  helix,  which  can  be  coiled 
about  the  shaft  of  the  gear,  for  it  can  evi- 
dently be  coiled  close  or  loose  without  affect- 
ing the  shape  of  its  teeth.  If  coiled  close, 
with  a  short  lead,  it  runs  nearly  at  right 
angles  to  the  shaft,  and  the  gear  approxi- 
mates to  the  spur  gear,  while  if  the  lead  is 
long  the  gear  approximates  to  the  screw. 

As  the  diameter  of  the  spiral  gear  increases, 
the  teeth  straighten,  and  when  the  diameter 
is  infinite  and  it  is  a  rack,  they  are  straight 
and  in  no  way  different  from  those  of  a  com- 
mon rack. 


104. — THEOKY   OF   SPIRAL   TOOTH   ACTION. 


The  Willis  theory  of  the  action  of  spiral 
teeth  is  the  one  generally  accepted,  but  it  is 
not  correct.  It  assumes  that  the  action  be- 
tween the  gears  is  upon  a  section  by  a  plane 
through  the  axis  of  the  gear  and  the  common 
normal  to  the  two  axes,  and  that  the  section 
of  the  two  gears  made  by  the  plane  act  to- 
gether like  a  rack  and  gear. 

When  the  axes  are  at  right  angles,  and  the 
spiral  angle  is  great,  this  theory  is  apparently 
correct,  the  error  being  practically  imper- 
ceptible, but,  as  the  axes  become  more  nearly 
parallel,  the  error  is  more  apparent,  until, 
when  they  are  parallel,  the  error  is  plain 
enough.  Willis  applied  his  theory  to  worms 
and  worm  gears,  on  axes  at  right  angles,  and 
evidently  did  not  consider  the  spiral  gear  in 
general. 

The  action  between  spiral  teeth  is  not  upon 
the  axial  section,  and  it  is  not  that  of  a  rack 
and  gear,  but  when  there  is  any  action  at  all 
it  is  upon  the  normal  spiral  section.  See  the 
AMERICAN  MACHINIST  for  May  19th,  1888. 

When  the  axes  are  parallel  the  normal 
spiral  sections,  as  well  as  the  sections  made 
by  a  plane  normal  to  the  axes,  are  conjugate, 
and  therefore  the  action  is  correct  and  along 
a  line  of  action.  The  action  is  also  continu- 
ous when  the  axes  intersect  and  the  gears  are 
bevel  gears. 


When,  however,  the  axes  are  askew,  the 
normal  spiral  sections  are  not  necessarily  con- 
jugate, for  they  coincide  only  on  one  line,  the 
common  normal  to  the  two  axes.  Therefore, 
there  is  no  continuous  tooth  contact,  except 
in  one  particular  case,  the  teeth  being  in  con- 
tact only  for  an  instant  as  they  pass  the 
normal. 

The  special  case  for  which  spiral  teeth  on 
askew  axes  have  continuous  tooth  contact,  is 
that  case  of  the  involute  tooth  when  the  base 
cylinders  are  tangent  and  the  gears  become 
spiraloidal  skew  bevel  gears.  See  (175)  and 
(176).  In  that  particular  case  the  teeth  have  a 
sliding  conjugate  action  on  each  other.  As 
the  spiraloidal  gear  is  fully  described  in  its 
place,  it  will  not  be  further  considered  here. 

This  theory  is  corroborated  by  experiment- 
al gears  made  for  the  Brown  &  Sharpe  Man- 
ufacturing Company,  for  whom  Mr.  O.  J. 
Beale,  to  whom  the  theory  of  the  spiral 
gear  is  much  indebted,  made  a  pair  of  theo- 
retically perfect  spiral  gears,  exactly  alike, 
with  a  spiral  angle  of  45°,  working  on  shafts 
at  right  angles,  and  of  such  a  large  size  that 
the  action  of  the  teeth  could  be  plainly  ob- 
served. See  Figs.  88  and  90. 

Beale's  gears  cannot  be  made  to  run  to- 
gether properly  at  any  shaft  distance,  but 
if  their  ends  are  brought  to  the  common 


54 


Spiral   Gearing. 


normal,  and  their  base  cylinders  are  in  con- 
tact, they  are  skew  bevel  gears  and  show  the 
action  required  by  Olivier's  theory. 

But,  although  the  action  of  spiral  gear 
teeth  is  intermittent,  and  their  contact  is  the- 
oretically perfect  at  one  instant  only,  when 


they  are  passing  the  common  normal,  they 
are  very  nearly  in  contact  all  the  time  and 
the  action  is  practically  perfect.  Spiral  teeth 
of  ordinary  sizes  work  together  with  a  re- 
markably smooth  action. 


105. — FORMATION    OF   THE    SPIRAL   TOOTH. 


As  the  spiral  rack  has  an  ordinary  straight 
tooth,  we  can  conveniently  derive  the  spiral 
tooth  in  general  from  it  by  a  method  that  is  a 
form  of  the  molding  method  of  (27)  for  spur 
gears. 

If  a  plane  is  moved  in  any  direction  upon  a 
cylinder  it  will  move  it,  as  if  by  friction, 
with  a  speed  that  depends  upon  the  direction 
of  the  motion.  If  we  imagine  the  same  re- 
sulting motion  between  the  plane  and  the 
pitch  cylinder,  and  assume  that  the  plane  is 
provided  with  hard  and  straight  teeth  run- 
ning in  any  direction,  it  will  mold  the  plastic 
substance  of  the  cylinder  and  form  spiral 
teeth  upon  it.  All  spiral  teeth  formed  by  the 
same  rack  will  have  normal  spiral  sections 
that  are  approximately  conjugate  to  each 
other,  and  they  will  work  together  inter- 
changeably. 

This  process  may  be  put  into  practical 
shape  by  a  modification  of  the  process  of  (28) 
for  spur  gears,  by  substituting  a  planing 
tooth  for  the  molding  rack  tooth.  The  tooth 
has  the  shape  of  the  normal  section  of  the 
rack,  and,  as  it  is  reciprocated  at  an  angle 
with  the  axis  of  the  gear  blank  being  shaped, 
both  the  tool  and  the  gear  blank  receive  the 
motion  of  the  plane  and  pitch  cylinder.  The 
cutting  face  of  the  tool  is  normal  to  the  direc- 
tion of  its  motion,  which  motion  is  tangent 
to  the  direction  of  the  tooth  spiral. 

The  linear  process  of  (29)  may  be  used,  the 
plane  of  Fig.  20  representing,  approximately, 
the  normal  spiral  section  of  the  gear.  Thus, 
if  the  planing  tool  or  the  equivalent  milling 
cutter  receives  a  motion  as  if  in  a  plane  roll- 
ing upon  the  base  cylinder,  the  involute  tooth 
will  be  produced. 


The  spiral  tooth  may  be  formed  by  the 
linear  planing  process  of  (29),  directly  ap- 
plied on  the  principle  that  the  spiral  tooth  is 
a  twisted  spur  tooth.  The  planing  tool  re- 
ceives a  planing  motion  in  the  direction  of 
the  axis  of  the  gear  blank,  and  both  tool  and 
blank  receive  the  feeding  rolling  motion  that 
would  produce  the  spur  tooth  of  the  section 
that  is  normal  to  the  axis.  In  addition,  the 
blank  receives  a  motion  of  rotation  while  the 
tool  moves,  that  is  repeated  for  every  troke 
of  the  tool.  The  cutting  edge  of  the  tool  is 
set  normal  to  the  axis  of  the  gear. 

The  spiral  tooth  may  also  be  formed  by  a 
tool  that  is  formed  to  the  true  shape  of  some 
section  of  the  tooth,  preferably  its  normal  sec- 
tion, and  which  is  guided  in  the  tooth  spiral. 
This  is  the  process  used  to  shape  a  worm,  the 
tool  being  guided  by  a  screw-cutting  lathe. 

The  process  generally  used  to  mill  the 
teeth  of  the  spiral  gear  is  the  equivalent  of 
the  operation  last  described.  The  milling 
cutter  is  shaped  to  the  normal  section  of 
the  tooth  space,  and  is  guided  in  the  tooth 
spiral  by  a  special  feeding  device  that  ro- 
tates the  blank  while  the  cutter  is  working 
in  it. 

Of  these  processes  the  planing  process  of 
(28)  is  the  best,  as  it  produces  the  tooth  with 
theoretical  perfection,  and  because  all  gears 
formed  with  the  same  tool  are  conjugate  and 
interchangeable.  But  the  screw-cutting  and 
milling  processes  are  most  in  use,  for  the 
reason  that  they  are  more  expeditious  and 
better  adapted  to  the  common  machine  tools, 
and  it  is  therefore  necessary  to  study  the 
shape  of  the  normal  section  of  the  tooth  with 
1  some  care. 


Spiral   Gearing. 


oo 


108. — THE  NORMAL  PITCH. 


The  real  pitch  of  the  spiral  gear  is  meas- 
ured on  a  section  that  is  normal  to  its  axis, 
and,  as  in  the  case  of  the  spur  gear,  it  is 
found  by  dividing  the  number  of  teeth  by 
the  pitch  diameter,  but  the  shape  of  the  tooth 
must  be  regulated  by  the  normal  pitch,  or 
pitch  of  its  normal  section. 

The  normal  pitch  is  found  by  dividing  the 


real  pitch  by  the  cosine  of  the  angle  made  by 
the  tooth  spiral  with  the  axis  of  the  gear. 
Thus,  if  the  pitch  is  8,  and  the  angle  is  45°, 
the  normal  pitch  is  8,  divided  by  .707,  or 
11.3. 

The  normal  circular  pitch  is  found  by  mul- 
tiplying the  real  circular  pitch  by  the  cosine 
of  the  spiral  angle. 


107. — THE    ADDENDUM. 

The  addendum  of  the  spiral  gear  should 
not  be  determined  by  its  real  pitch,  but  by 
its  normal  pitch,  for  it  is  then  usually  possi- 
ble to  mill  the  tooth  with  a  milling  cutter 
that  is  made  for  a  standard  spur  gear.  A 
gear  of  8  pitch  and  45°  angle  should  have  an 
1 


addendum  of 


11.3 


=  .089". 


If  the  addendum  is  determined  by  the  true 
pitch  when  the  angle  is  considerable,  the 
tooth  will  be  long  and  thin.  Fig.  86  shows 
the  normal  pitch  section  of  a  rack  to  run 
with  a  pinion  of  45°  angle,  while  Fig.  87 
shows  the  true  pitch  of  the  same  rack.  Fig. 
88  also  shows  the  true  pitch  of  the  pinion, 
and,  although  the  tooth  appears  to  be  stunted, 
it  is  really  of  the  standard  shape. 


Spiral  Rack, 
Fig.  87. 


Beetle's  Experimental  Gears. 
Fig.  88. 


108. — THE  AXIAL  PITCH. 


The  section  of  the  spiral  gear  by  a  plane 
through  the  axis  is  that  of  a  rack,  and  the 
axial  pitch,  or  pitch  of  the  rack,  is  found  by 
dividing  the  true  pitch  by  the  tangent  of  the 


spiral  angle.  Thus,  if  the  angle  is  45°,  the 
axial  pitch  is  the  same  as  the  true  pitch,  but 
the  axial  pitch  of  a  70°  spiral  tooth  is  but  .364 
of  the  true  pitch. 


109. — SHAPING  THE  TOOL. 


When  the  spiral  gear  is  cut  in  a  milling 
machine,  or  turned  in  a  lathe,  it  is  necessary 
to  give  the  tool  the  shape  of  the  normal  sec- 
tion of  the  tooth  to  be  cut,  and  this  is  most 
readily  accomplished  by  shaping  it  for  the 
spur  gear  that  most  nearly  coincides  with  that 
normal  section. 


The  number  of  teeth  in  the  gear  that  is 
osculatory  to  the  normal  spiral,  and  therefore 
most  nearly  coincides  with  it,  is  found  by 
dividing  the  actual  number  of  teeth  in  the 
gear  by  the  third  power  of  the  cosine  of  the 
spiral  anple. 

For  example,  if  we  are  to  cut  a  gear  of  4" 


56 


Spiral   Gearing. 


diameter,  6  pitch,  and  24  teeth,  at  a  spiral 
angle  of  45°,  the  cutter  should  be  shaped  to 

24  24 

cut  a  spur  gear  of  =  — —  =  69  teeth 

of  -=^  =  8.5  pitch.    If  the  gear  has  28  teeth 

of  4  pitch,  and  an  angle  of  10°,  the  equiva- 
lent spur  gear  has  29  teeth  of  4.08  pitch,  as 
the  gear  varies  but  little  from  a  spur  gear. 
If  the  gear  is  of  5  pitch,  and  15  teeth,  with 
an  angle  of  80°,  the  equivalent  spur  gear  has 
2,830  teeth  of  28.7  pitch,  and  in  general, 
when  the  gear  has  a  great  angle  it  is  a 


worm,  the  section  is  practically  that  of  a  rack. 
Care  must  be  taken,  when  the  gear  is  a 
screw,  and  is  turned  in  the  lathe,  that  the 
tool  should  be  set  with  its  cutting  edge  nor- 
mal to  the  thread  of  the  screw,  if  it  is  shaped 
by  the  above  rule.  If  the  tool  is  set  in  the 
axial  section  of  the  screw,  and  it  generally  is, 
it  should  be  shaped  to  the  axial  section  of  the 
worm,  and  have  the  axial  pitch  and  adden- 
dum. But  when  the  lead  of  the  thread  of 
the  screw  is  small  compared  with  its  diam- 
eter the  difference  between  the  normal  and 
axial  sections  is  not  noticeable. 


110. — VELOCITY   RATIO   OP   SPIRAL   GEARS. 


The  spiral  gear  does  not  follow  the  well- 
known  rule  of  spur  gears,  that  the  velocities 
in  revolutions  in  a  given  time  are  inversely 
proportional  to  the  pitch  diameters,  but  re- 
quires that  ratio  to  be  multiplied  by  the  ratio 
of  the  cosines  of  the  spiral  angles. 

In  the  formula 

v  I)     cos.  A 

~V       ~~    d      cos.  a 

D  and  d  are  the  diameters  of  the  gears,  A 
and  a  are  their  spiral  angles,  and  V  and  « 
are  their  velocities  in  revolutions. 

If  the  angles  are  equal,  the  velocity  ratio 
is  the  same  as  for  spur  gears  of  the  same 
diameters.  Fig.  88  shows  a  pair  of  gears  B 
and  G  that  are  of  the  same  size  and  have  the 
same  angle  in  opposite  directions,  requiring 
the  shafts  to  be  parallel.  See  also  Fig.  89. 
The  pair  of  gears  A  and  B  are  exactly  alike, 
with  equal  angles  in  the  same  direction,  re- 
quiring the  shafts  to  be  at  an  angle  equal  to 


Fig.  SO 
Spiral  Spur  Gears 


Equa 
Gears 


Fig.  00 


twice  the  spiral  angle.  See  also  Fig.  90. 
The  statement  that  like  spiral  gears  will  not 
run  together  is  founded  on  the  Willis  theory 
of  spiral  gear  contact,  and  is  wrong. 


111. — SPIRAL  WORM  AND   GEAR. 


When  the  shafts  are  at  right  angles,  and 
the  angle  on  one  is  so  great  that  it  is  a  screw, 
the  combination  is  known  as  a  worm  gear 
and  worm,  Figs.  91  and  92,  and  is  much  used 
for  obtaining  slow  and  powerful  motions. 
It  is  also  too  much  used  for  wasting  power 
and  wearing  itself  out,  for  its  friction  is  very 
great  and  consumes  from  one-quarter  to 
two-thirds  of  thd  power  received . 

When  the  screw  has  a  single  thread,  the 


velocity  ratio  is  simply  the  number  of  teeth 
in  the  gear,  and  if  th  re  are  two  or  three 
threads  it  must  be  modified  accordingly. 

The  spiral  worm  is  adjustable  in  its  gear 
both  laterally  and  longitudinally,  so  that  it 
will  change  its  position  as  required  by  wear 
in  the  shaft  bearings. 

It  is  an  excellent  substitute  for  the  hobbed 
worm  and  gear,  and  in  most  cases  will  serve 
practical  purposes  quite  as  well. 


Worm    Gearing. 


57 


Spiral  Worm  Gear  and  Worm, 
Fig.  91. 


Fig.  92 


Worm  Gears 


112. — EFFICIENCY  OF  SPIRAL  AND  WORM  GEARING. 


Unless  the  shafts  are  parallel  the  teeth  of  a 
pair  of  spiral  gears  are  moving  in  different 
directions,  and  therefore  they  cannot  pass 
each  other  without  sliding  on  each  other  an 
amount  that  increases  rapidly  with  the  angle 
of  divergence  of  the  directions  of  motion, 
that  is,  the  shaft  angle. 

This  sliding  action  creates  friction  and 
tends  to  wear  the  teeth,  and  to  a  very  much 
greater  extent  than  is  generally  supposed. 
The  friction  is  so  great,  in  fact,  that  such 
gears,  particularly  worm  gears,  should  be 
used  only  for  conveying  light  powers.  They 
are  extensively  used,  or  rather  misused,  for 
driving  elevators,  and  are  even  found  in  mill- 
ing machines,  gear  cutters,  planers,  and 
similar  places,  in  evident  ignorance  that  they 
waste  from  a  quarter  to  two-thirds  of  the 
power  received. 

The  most  extensive  experiments  on  the 
efficiency  of  spiral  and  worm  gears  ever  made 
were  made  by  Wm.  Sellers  &  Co.,  and  they 
may  be  found  described  in  great  detail  in  a 
paper  by  Wilfred  Lewis  in  the  Transactions 
of  the  American  Society  of  Mechanical  En- 
gineers, vol.  vii.  Space  will  not  permit  ex- 
tensive quotations  from  this  valuable  paper, 
but  the  general  result  of  the  experiments  is 


shown  by  the  diagram,  Fig.  93.  The  diagram 
shows  that  a  common  cast-iron  spur  gear  and 
pinion  on  parallel  shafts  have  an  efficiency  of 
from  ninety  to  ninety-nine  per  cent.,  accord- 
ing to  the  speed  at  which  they  are  working  ; 
that  a  spiral  pinion  of  45°,  angle  working  in 
a  spur  gear,  with  shafts  at  45°,  has  an  effi- 
ciency of  from  81  to  97  per  cent. ;  that 
the  efficiency  decreases  as  the  angle  of  the 
shafts  increases,  until,  for  a  worm  of  a  spiral 
angle  of  5°,  at  a  shaft  angle  of  85°,  it  goes 
as  low  as  34,  and  does  not  rise  higher  than  77 
per  cent.  This  includes  the  waste  of  power 
at  the  shaft  bearings  as  well  as  that  at  the 
teeth  of  the  gears.  The  efficiency  is  lowest 
for  slow  speeds,  and  rises  with  the  speed. 
The  diagram  may  be  relied  upon  to  give  its 
true  value,  under  ordinary  conditions,  within 
five  per  cent. 

The  same  experiments  developed  the  fact 
that  the  velocity  of  the  sliding  motion  of  the 
cast-iron  teeth  on  each  other  should  not  be 
over  two  hundred  feet  per  minute  in  contin- 
uous service,  to  avoid  cutting  of  the  surfaces. 
It  may  be  assumed  that  the  efficiency  will  be 
higher  when  the  worm  is  of  steel,  particu- 
larly when  the  gear  is  of  bronze. 

Diagram,  Fig.  94,  shows  the  result  of  simi- 


58 


Worm    Gearing. 


Velocity  at  Pitch  Xine  in  feet  per  minute, 

o  ,,^,,22    2    §        S     §  g  S 


-feq 


£ 


-S;.,  P».  20" 


<M  co      r»«*o-oi-aoaioc*»oo 

Sellers'   Experiments 

Fig.  93. 


lar  experiments  by  Prof.  Thurston,  with  a 
worm  of  6"  diameter  and  one  inch  circular 
pitch  running  in  a  gear  of  16"  diameter,  both 
cast-iron. 

It  is  to  be  observed  that  it  is  the  shaft  angle, 
and  not  the  angle  of  the  spiral,  that  deter- 
mines the  efficiency.  A  pair  of  spiral  gears 
on  parallel  shafts  are  practically  as  efficient 
as  gears  with  straight  teeth. 

The  great  friction  of  worm  gearing  is  of 
advantage  for  one  purpose,  and  for  one  only, 
to  secure  safety  and  prevent  undesired  mo- 
tion of  the  gears.  The  worm  of  Fig.  97  will 
easily  move  the  gear,  but  the  gear  must  be 
moved  with  great  force  to  start  the  worm. 
When  the  angle  of  the  worm  is  as  small  as 
the  "angle  of  repose"  for  the  metals  in 
contact,  it  is  impossible  for  the  gear  to  drive 
the  worm.  This  may  be  an  excuse  for  the  use 
of  the  worm  gear  in  elevators,  but  it  would 
seem  that  the  safety  of  the  cage  should  de- 


Revolutions  of  Driver  per  minute 
50     100    150     200     250    300     350    400 


Yale  &  Totvue  Experiments 

Fig.  94. 


pend  on  devices  attached  to  the  cage  itself, 
rather  than  to  the  hoisting  machinery  or 
other  distant  part. 

Unless  the  friction  of  the  gears  must  be 
depended  upon  for  safety,  the  worm  gear 
should  be  used  only  for  purposes  of  adjust- 
ment, or  when  speed  must  be  greatly  reduced 
or  power  increased  within  a  small  compass, 
and  not  for  conveying  power. 


Worm    Gearing. 


113. — THRUST  OF   SPIRAL  TEETH. 


The  oblique  action  of  the  teeth  of 
spiral  gears  on  each  other  tends  to  throw 
the  gears  bodily  in  the  direction  of  their 
axes,  and  this  tendency  creates  a  thrust 
that  must  be  opposed  by  thrust  bear- 
ings. The  end  pressure  on  the  shaft  of 
a  worm  is  greater  than  that  exerted  on 
the  teeth  of  the  worm  gear  it  is  driving. 

When  the  shafts  are  parallel  the 
thrust  may  be  completely  avoided  by 
the  use  of  double  spiral  or  "herring- 
bone "  teeth,  Fig.  82  or  83,  which  act  in 
opposite  directions,  and  neutralize  each 
other. 

When  the  shafts  are  at  right  angles 
the  thrust  may  be  neutralized  by  op- 
posing a  second  gear  in  the  manner 
shown  in  section  by  Fig.  95.    The  two 
worms  with  opposite  spirals  run  in  two  spiral 
worm  gears  that  also  work  with  each  other, 
and,  as  the  pressure  on  one  gear  is  opposite 
that  on  the  other,  there  is  no  thrust  on  the 
shaft.     When  this  combination  is  made  with 
worm  gears  having  concave  teeth,  the  teeth 
can  bear  only  at  their  ends. 


Arrangement  to  avoid  thrust 

Fig.  95. 


When  the  thrust   cannot  be   avoided   it 

should  be  taken  by  a  roller  bearing,  rather 

j  than  by  the  common  collar  bearing.    The 

I  diagram,  Fig.  94,  shows  the  superior  efficiency 

of  the  roller  bearing  as  compared  with  the 

collar  bearing,  the  gain  being  from  ten  to 

•  twenty  per  cent. 


114. — THE  HOBBED  OR  CONCAVE  WORM  GEAR. 


If  a  spiral  gear  is  made  of  steel,  provided 
with  cutting  edges  by  making  slots  across  its 
teeth,  and  hardened,  it  will  be  a  practical 
cutting  tool  called  a  spiral  milling  cutter  or 
hob.  Fig.  96  shows  a  spiral  milling  cutter, 
having  a  great  spiral  angle,  and  therefore 
called  a  worm. 

If  this  cutting  spiral  gear  is  mounted  in 
connection  with  an  uncut  blank  so  that  both 
are  rotated  with  the  proper  speeds,  and  the 
shafts  of  the  two  gears  are  gradually  brought 
together  while  they  are  revolving,  the  edge 
of  the  blank  will  be  formed  with  concave 
teeth  that  curve  upwards  about  the  sides  of 
the  cutting  gear.  If  the  hob  is  then  replaced 
with  a  spiral  gear  that  is  a  duplicate  of  it,  ex- 
cept that  it  has  no  cutting  teeth,  we  shall 
have  the  familiar  worm  and  worm  gear  of 
Fig.  97. 

The  principle  of  the  concave  gear  applies 
to  any  pair  of  spiral  gears,  on  shafts  at  any 


Concave  Worm  Gear  and  Worm. 
Fig.  97. 


Worm    Gearing. 


angle,  but  in  practice  it  is  confined  to  the 
worm  and  gear  on  shafts  at  right  angles. 

The  nature  of  the  contact  between  the 
worm  and  the  concaved  worm  gear  has  not 
yet  been  definitely  determined,  but  there  is 
no  reason  to  suppose  that  it  is  different  from 
that  between  plain  spiral  teeth,  a  point  con- 
tact on  the  normal  spiral,  but  it  is  probably 
continuous.  It  is  certain,  however,  that  the 
contact  is  considerably  closer,  more  nearly 
resembling  surface  contact,  and  being  sur- 
face contact  when  the  diameter  of  the  gear  is 
infinite. 

The  worm  is  adjustable  in  the  concaved 
teeth  of  the  gear  in  the  direction  of  its  axis, 


A  Hob. 


and  will  change  its  position  as  required  by 
the  wear  of  the  thrust  bearing.  It  is  not  ad- 
justable laterally. 


115. — HOBBING  THE  WORM  GEAR. 


When  the  hob  is  provided  it  is  a  simple 
matter  to  cut  the  gear.  The  gear  is  generally 
provided  with  the  desired  number  of  notches 
in  its  edge,  that  are  deep  enough  to  receive 
the  points  of  the  teeth  of  the  hob,  and 
the  hob  will  then  pull  it  around  as  it 
revolves. 

It  is  a  too  common  practice  to  make  the 
hob  do  its  own  nicking,  for,  if  it  is  forced 
into  the  face  of  the  gear  as  it  revolves,  it  will 
pull  it  around  by  catching  its  last  teeth  in 
the  nicks  made  by  the  first. 

If  luck  is  good  these  nicks  will  run  into 
each  other,  and  the  gear  will  be  cut  with 
teeth  that  appear  to  be  correct,  but,  as  the 
outside  diameter  js  greater  than  the  pitch 


diameter,  there  will  be  one,  two,  or  three 
teeth  too  many.  The  teeth  of  the  finished 
gear  are  therefore  smaller  than  those  of  the 
worm  by  an  amount  that  is  ruinous  if  the  gear 
is  small,  although  it  is  not  noticeable  when 
the  diameter  is  large.  If  there  are  12  teeth 
where  there  should  be  but  10,  each  tooth  will 
be  too  small  by  two-twelfths  of  itself;  but  if 
there  are  102  teeth  where  there  should  be  but 
100,  each  tooth  is  too  small  by  but  two- 
hundredths  of  itself.  This  handy  makeshift 
process  will  do  very  well  on  large  ge^rs,  but 
not  on  small  ones,  unless  the  worm  to  run  in 
the  gear  is  made  to  fit  the  tooth,  with  a  tooth 
that  is  smaller,  and  lead  that  is  shorter  than 
that  of  the  hob. 


116.— INVOLUTE    WORM    TEETH. 


Worms  are  generally  cut  in  the  lathe,  and  ! 
as   a   straight-sided    tooth    is    most    easily 
formed,    the    involute    tooth    is    generally 
adopted. 

Strictly,  the  form  of  the  tool  should  be 
that  of  the  normal  section  of  the  thread,  and 
it  should  always  be  set  in  the  lathe  with  its 
cutting  face  at  right  angles  to  the  thread.  | 


But  custom  and  convenience  allow  the  tooth 
to  have  S'raight  sides,  and  to  be  set  with  its 
face  parallel  with  the  axis  of  the  worm,  and 
the  real  difference  is  not  generally  notice- 
able. 

The  standard  tool  has  its  sides  inclined  at 
an  angln  of  30°,  and  has  a  length  and  a  width 
dependent  upon  the  pitch. 


117. — INTERFERENCE   OF   INVOLUTE   WORM   TEETH. 

There  is  one  difficulty  that  is  seldom  recog- 1  pected,  and  that  is  interference.  The  teeth 
nized,'  but  which  must  be  carefully  guarded  of  worm  gears  will  interfere  with  each  other 
against  if  properly  running  gears  are  ex- !  when  the  conditions  are  right  for  interference. 


Worm 


Gearing'. 


just  as  spur  involute  tee.th  will  interfere,  as 
shown  by  Fig.  36.     Fig.  98  shows  the  gear 
that  would  be  formed  by  the  usual  process. 
The  difficulty  can  be  remedied  by  rounding 


over  the  tops  of  the  teeth  of  the  hob  and 
worm,  as  described  in  (55). 

It  is  also  a  simple  matter  to  avoid  the  inter- 
ference by  enlarging  the  outside  diameter  of 


Interfering  Worm, 
Fig.  98. 


Interference  Avoided. 
Fig.  99. 


\ 


Involute  worm  anil  gear 
twenty-one  or  more  teeth 

Fig.  100. 


Worm    Gearing. 


the  worm  gear.  If,  as  shown  by  Fig.  99,  the 
tooth  has  but  a  short  flank,  or  none  at  all, 
and  the  addendum  of  the  gear  is  about  twice 
that  by  the  usual  rule,  the  action  will  be  con- 
fined to  the  face  of  the  gear  and  the  flank  of 
the  worm,  and  there  can  be  no  interference. 
By  adopting  an  obliquity  greater  than  15°, 
interference  can  be  avoided  without  changing 
the  addendum. 

This  method  has  the  advantage  that  the 
same  straight-sided  worm  and  hob  can  be 
used  for  small  gears  as  for  large  ones,  and 
the  disadvantage  that  the  action  is  confined 
to  the  approach  and  subject  to  greater  fric- 
tion (48). 

When  the  standard  30°  tool  is  used,  all 
gears  of  26  teeth,  or  smaller,  should  be  made 
in  this  way,  but  the  correction  is  not  strictly 
necessary  for  gears  of  more  than  20  teeth, 
unless  particularly  nice  work  is  required. 

Fig.  100  shows  the  proper  construction  of  a 
gear  of  21  or  more  teeth,  and  Fig.  101  shows 
that  of  a  gear  of  less  than  21  teeth.  In  the 
former  case,  the  teeth  of  the  worm  should  be 
limited  by  the  limit  line  II,  but  the  interfer- 
ence for  21  to  25  teeth  is  not  noticeable. 


Draw  worm  teeth  straight 
Draw  gear  teeth  by  points  (57) 


Involute  worm 
for  twenty^  or 


118. — CLEARANCE   OF  WORM   TEETH. 


There  is  another  practical  point  that  is  sel- 
dom recognized,  and  that  is  that  worm  teeth 
should  have  clearance  (40),  for  there  is  no 
reason  for  clearing  spur  teeth  that  will  not 
apply  quite  as  well  to  any  other  kind. 

The  clearance  is  easily  obtained  by  making 
the  tooth  of  the  hob  a  little  longer  than  that 
of  the  worm,  as  shown  by  the  tooth  a  of  Fig. 


'100.  For  the  same  reason  the  hob  should 
have  no  clearance  at  the  bottom  of  its  thread, 
so  that  the  tops  of  the  gear  teeth  will  be 
formed  of  the  proper  length.  The  custom 
of  making  the  hob  and  worm  of  exactly  the 
same  diameters  will  apply  only  when  the 
worm  "bottoms"  in  the  gear  and  the  gear 
bottoms  in  the  worm. 


119. — CIRCULAR  PITCH    WORM   TEETH. 

The  old  and  clumsy  circular  pitch  system  \  were  the  same  as  those  of  common  spur  gears 
is  in  universal  use  for  worm  teeth,  for  the  |  and  racks  on  the  circular  pitch  system.    The 


reason  that  worms  are  generally  made  in  the 
lathe,  and  lathes  are  never  provided  with  the 
proper  change  gears  for  cutting  diametral 
pitches.  The  error  is  so  firmly  rooted  that  it 
is  useless  to  attempt  to  dislodge  it. 

It  is  therefore  necessary  to  figure  the  diam- 
eters of  worm  gears  as  if  their  throat  sections 


table  of  diameters  (35)  will  be  of  great  assist- 
ance. 

One  great  objection  to  the  use  of  the  circu- 
lar pitch  system  for  spur  gears  does  not  ap- 
ply to  worm  gears,  that  the  center  distance 
between  the  shafts  will  always  be  an  incon- 
venient fraction,  unless  the  pitch  is  as  incon- 


Diametral    Worm    Gearing. 


63 


venient.  The  worm  can  be  made  of  any 
diameter,  and  can  therefore  be  made  to  suit 
the  pitch  diameter  of  the  gear  and  the  center 
distance  at  the  same  time. 

The  sides  of  the  tool  for  circular  pitches 
should  come  together  at  an  angle  of  thirty 
degrees,  and  the  width  of  the  point,  as  well 
as  the  depth  to  be  cut  in  the  worm  or  in  the 
hob,  should  be  taken  from  the  following 
table.  The  diameter  of  the  hob  should  be 
greater  than  that  of  the  worm  by  the  "in- 
crease" given. 

Make  the  tool  with  the  proper  width  at  the 
point  to  thread  the  worm,  and  then,  after 
making  the  worm,  grind  off  half  the  "  in- 
crease" from  the  length  of  the  tool,  and  use 
it  to  thread  the  hob. 


TABLE  FOR  CIRCULAR  PITCH   WORM  TOOLS. 


Circular  ]5itch 

2 

13£ 

lUi 

\\t 

1^6 

Point  of  hob  tool  
Point  of  worm  tool.  .  . 
Depth  of  cut  in  worm 
or  hob  
Increase  

.644 
.620 

1.416 
.I6b 

.564 
.542 

1.240 
.146 

.483 
.466 

1.062 
1.249 

.402 
.388 

.886 
.104 

.362 
.349 

.797 
.094 

Circular  pitch  
Point  of  hob  tool  

1 

.322 

% 
.282 

K 

.241 

% 
.201 

H 
161 

Point  of  worm  tool.  . 
Depth  of  cut  in  worm 
or  hob 

.310 
708 

.271 
620 

.233 
531 

.194 
443 

.155 
354 

Increase  

.Ob3 

.073 

.062 

.052 

^042 

Circular  pitch. 

j. 

% 

JL 

YA. 

JL 

Point  of  hob  tool  
Point  of  worm  tool  .  . 
Depth  of  cut  in  worm 
or  hob  

.141 
.135 

310 

.121 
.116 

265 

100 

.097 

222 

.080 
.078 

177 

.060 
.058 

133 

Increase  

.036 

.031 

.026 

.0^1 

.016 

120. — DIAMETRAL   PITCH   WORM   TEETH. 


If  the  proper  change  gears  are  provided,  it 
is  as  easy  to  cut  diametral  pitch  worm  teeth 
as  any.  The  proper  gears  can  always  be 
easily  calculated  by  the  rule  that  the  screw 
gear  is  to  the  stiyl  gear  as  twenty-two  times  the 
pitch  of  the  lead  screw  of  the  lathe  is  to  seven 
times  the  diametral  pitch  of  the  worm  to  be  cut. 

For  example,  it  is  required  to  cut  a  worm 
of  twelve  diametral  pitch,  on  a  lathe  having 
a  leading  screw  cut  six  to  the  inch.     We  have 
screw  gear  _  22  X    6  _   11 
stud  gear          7  X  12  ~    7~' 
and  any  change  gears  in  the  proportion  of  11 
and  7  will  answer  the  purpose  with  an  error 
1 


of 


of  an  inch  to  the  thread  of  the  worm. 


10,000 

If  22  and  7  give  inconvenient  numbers  of 
teeth,  the  numbers  69  and  22  can  be  used 
with  sufficient  accuracy,  and  47  and  15,  or 
even  25  and  8  may  do  in  some  cases. 

To  save  calculation  and  study,  the  table  of 
change  gears  for  diametral  pitches  is  provided, 
and  it  will  give  the  proportion  of  screw  gear 
to'slud  gear  to  be  used  for  all  ordinary  cases. 

The  pair  on  ihe  left  will  give  the  proper 
pitch  within  less  than  a  thousandth  of  an 
inch,  and  that  on  the  riglit  will  serve  with  an 
error  always  less  than  a  hundredth  of  an  inch, 
and  sometimes  less  than  two  or  three  thou- 
sandths of  an  inch. 

Having  the  change  gears,  figure  the  pitch 


diameter  of  the  gear  as  if  the  throat  section 
is  a  spur  gear  on  the  diametral  pitch  system. 
The  sides  of  the  tool  should  come  together 
at  an  angle  of  thirty  degrees,  and  the  width 
of  the  point  of  the  tool,  as  well  as  the  depth 
to  be  cut  in  the  worm  or  in  the  hob,  should  be 
taken  from  the  following  table.  The  diame- 
ter of  the  hob  should  be  greater  than  that  of 
the  worm  by  the  "increase"  given. 

TABLE  FOR  DIAMETRAL  PITCH  WORM  TOOLS. 


Diametral  pitch.  

Point  of  hob  tool  
Point  of  worm  tool  
Depth  of  cut  in  worm  or 
hob 

1 

1.035 
.968 

2.125 

2 

.517 
.484 

1.063 

3 

.345 
.323 

.708 

4 

.258 
.242 

532 

Increase  

.250 

.125 

.083 

.063 

Diametral  pitch  

Pointof  hob  tool  
Point  of  worm  tool  .  . 
Depth  of  cut  in  worm  or 
hob                               •* 

5 

.207 
.194 

.425 

6 

.173 
.162 

.354 

7 

.148 
.138 

304 

8 

.129 
1-A 

.266 

Increase  .... 

.050 

.042 

.036 

.032 

Diametral  pitch  

Point  of  hob  tool  
Point  of  worm  tool  
Depth  of  cut  in  worm  or 
hub 

10 

.104 
.097 

213 

12 

.086 
.081 

.177 

14 

.074 
.069 

.152 

16 

.065 
.060 

.138 

Increase  

.025 

.021 

.018 

.016 

Make  the  tool  with  the  proper  width  at 
the  point  to  thread  the  worm;  and  then, 
after  making  the  worm,  grind  off  half  the 
"increase  "  from  the  length  of  the  tool,  and 
use  it  to  thread  the  hob. 


64 


Diametral    Worm    Gearing. 


3 

4 

. 

5 

3 

0 

0) 
JD 

6 

O 

JC 

7 

a 

8 

"as 

0) 
ctf 

10 

5 

12 

14 
16 


Pitch  of  Leading:  Screw. 

4567 


10 


44  23 
2l'  11 

22 
7  ' 

88  46 
2l'  11 

110 
21 

44  69 
7  'll 

21 
3  ' 

176  92 
21  '  11 

220 
21  ' 

11 

7  ' 

33 

14' 

22 

7  ' 

55 
H' 

33 

7  ' 

11 
2  ' 

44 

7  ' 

55 

7  ' 

44  5 
35'  4 

66  15 
35*  8 

88  5 
,35'  2 

22  25 

7  '  8 

182  15 
35  '  4 

22  35 
H'  ~S 

176  5 
35  '  1 

44  25 

7  '  4 

22 

11  30 

44  40 

55  50 

22  60 

11  70 

88  80 

110  100 

21- 

7  19 

21  19 

21  19 

7  19 

3  19 

21  19 

21   19 

44  9 

66  27 

88  9 

110  9 

132  27 

22  63 

176  18 

220  9 

49  10 

49  20 

49  5 

49   4 

49  10 

7  20 

49   5 

49   2 

11  4 

33  6 

11  8 

55  2 

33  12 

11  14 

22  16 

55  4 

14  5 

28  5 

7   5 

28  1 

14  5 

4   5 

7   5 

14  1 

22  5 

33  15 

44  5 

11  25 

66  15 

11  35 

88  5 

22  25 

35  8 

35  16 

35  4 

7  16 

35  8 

5  16 

35  4 

7  8 

11  10 

11  15 

22  20 

55  25 

11  30 

11  35 

44  40 

55  50 

21  19 

14  19 

21  19 

42  19 

7  19 

6  19 

21  19 

21  19 

22  4 

33  2 

44  8 

55  10 

66  4 

11  14 

88  16 

110  20 

49  9 

49'  3 

49  9 

49  9 

49  3 

7   9 

49  9 

.49   9 

11  2 

33  3 

22  4 

55  1 

33  6 

77  7 

11  8 

55  2 

28  5 

56  5 

28  5 

56  1 

28  5 

56  5 

7   5 

28  1 

Exact  numbers  on  the  left.     Approximate  on  the  right. 
TABLE  OP  CHANGE  GEARS  FOR  DIAMETRAL  PITCH  WORMS. 


121. — WIDTH  OF  WORM  GEAR  FACE. 

The   bearing    between   the    tooth    of  the  '     The  length  of  the  worm  need  be  no  more 


worm  and  that  of  the  gear  is  near  the  center 
of  the  gear,  and  it  is  quite  small  (104).  It  is, 
therefore,  useless  to  make  the  gear  with  a 
wide  face.  If  the  face  is  half  the  diameter 
of  the  worm  it  will  have  all  the  bearing  that 
can  be  obtained,  and  any  extra  width  will 
simply  add  to  the  weight  and  cost  of  the  gear. 


than  three  times  the  circular  pitch,  for  there 
are  seldom  more  than  two  teeth  in  contact  at 
once.  If,  however,  the  worm  is  made  lorg, 
it  can  be  shifted  when  it  becomes  worn,  so 
as  to  bring  fresh  teeth  into  working  position. 
This  provision  is  wise,  for  the  reason  that  the 
worm  is  always  worn  more  than  the  gear. 


122. — THE   HINDLEY   WORM   AND 


If  the  cutting  hob  and  the  worm  is  shaped 
by  the  tool  a,  and  the  process  indicated  by 
Fig.  102,  the  resulting  pair  of  gears  is  known 


It  is  commonly  but  erroneously  stated 
that  this  worm  fits  and  fills  its  gear  on  the 
axial  section,  the  section  that  is  made  by  a 


as  the  Hindley  worm  and  gear.     The  worm  j  plane  through  the  axis  of  the   worm  and 
is  often  called  the  "hour-glass"  worm.  I  normal  to  the  axis  of  the  gear.     It  has  even 


Hindi ev    Worm    Gearing'. 


(55 


been  stated  that  the  contact  is  between  sur- 
faces, the  worm  tilling  the  whole  gear  tooth. 

The  real  contact  is  not  yet  certain,  but  it  is 
certain  that  it  is  not  a  surface  contact.  It  is 
also  certain  that  it  is  on  the  normal  and  not 
on  the  axial  section,  and  that  the  Hindley 
worm  hob  will  not  cut  a  tooth  that  will  till 
any  section  of  it.  The  contact  may  be  linear, 
along  some  line  of  no  great  length,  but  it  is 
probably  a  point  contact  on  the  normal  sec- 
tion. The  order  of  the  contact  is  certainly 
very  close,  resembling  that  of  two  surfaces. 

The  worm  is  limited  in  length,  for  the 
sides  of  the  teeth  cannot  slant  inward  from 
the  normal  to  the  axis.  The  end  tooth  m  in 
Fig.  102  cannot  be  used,  for  it  will  destroy 
the  teeth  of  the  gear  as  it  is  fed  towards 
this  axis  in  the  operation  of  hobbing. 

It  has  the  one  great  defect  that  it  is  not 
adjustable  in  any  direction,  and,  therefore, 
cannot  change  its  position  when  the  shaft 


The  Hindley 
Worm  Gear 


bearings  wear,  unless  it  is  itself  worn  the 
same  amount.  It  is  doubtful  if  this  form  of 
gearing  has  any  advantage  over  the  plain 
spiral  gearing,  except  when  new  and  in  per- 
fect adjustment. 


123. — THE   PIN   WORM  AND   GEAR. 

If  the  hob  and  the  worm  are  shaped  by   Fig.   102,   the  gearing  produced  will  have 
the   pin-shaped  revolving  milling  tool  b  of  !  linear  bearing  between  the  teeth. 

The  action  will  be  the  same  as  between  a 
series  of  pin  teeth  like  the  milling  tool,  each 
pin  being  in  the  axial  section  of  the  worm, 
but  having  a  linear  bearing  on  the  normal 
section  of  its  teeth. 

This  form    of    gearing,    which  is  a 
modification    of    the    Hindley    form, 
may   take   the  shape  of  pin  gearing, 
the  teeth   being   round    pins   like  the 
milling  tool.     If  the  pins  are  mounted 
on  studs,  so  as  to  revolve,  a  roller  pin  worm 
gear  will  be  produced. 

Fig.  103  shows  a  form  of  roller  pin 
gearing  in  which  the  pins  have  been  en- 


2'in  worm-  year. 

Fig.  103. 


larged. 


124. — THE   WHIT  WORTH    HOBBING    MACHINE. 


When  the  amount  of  work  to  be  done  will 
warrant  the  use  of  a  special  machine,  the 
hobbing  machine  of  Sir  Joseph  Whit  worth 
may  be  used.  It  was  invented  in  1835,  and 
has  not  been  materially  improved  since  then, 


although  there  are  numerous  patents  relating 
to  it.  The  worm  gear  to  be  hobbed  is  fixed 
upon  the  same  spindle  with  a  master  worm- 
wheel.  A  driving  worm  runs  in  the  master 
wheel,  and  it  is  connected  by  a  train  of  gear- 


66 


Hobbing  Machines. 


ing  with  a  hob  that  is  so  mounted  on  a 
carriage  that  it  caii  be  fed  towards  the  gear 
blank.  The  hob  is  slowly  forced  into  the 
blank,  while  both  are  revolving  with  the 
proper  speeds,  and  the  gear  is  cut  without 
the  assistance  of  previously  made  hicks.  See 
British  patent  6,850  of  1835. 


loc 

Spiral  and  Spur  Gear. 


125. — THE  CONJUGATOR. 


This  is  a  machine  for  cutting  spur  or  spiral 
gears  by  means  of  a  hob,  and  its  principle  is 
an  extension  of  that  of  the  Whitworth  worm 
gear  hobbing  machine. 

If,  when  the  hob  in  the  Whitworth  ma- 


Conjugator.    Elevation 

Fig.104. 


flan 

Tig.  105. 


chine  has  reached  the  full  depth  of  the  tooth, 
it  receives  a  new  .motion  in  the  direction 
of  the  tangent  to  its  pitch  spiral,  it  will 
continue  the  tooth  to  the  edge  of  the  gear, 
and  form  the  plain  spiral  gear  of  Fig.  91 . 

Fig.  104  is  an  elevation  of  the  machine,  and 
Fig.  105  is  a  plan.  The  hob  h  is  mounted 
upon  an  arbor  that  is  connected  by  a  train  of 
gearing  with  the  spindle  *  that  carries  the 
blank  gear  g  to  be  cut,  so  that  the  hob  and 
blank  revolve  together  with  any  definite 
proportionate  speed.  The  hob  is  carried 
upon  a  carriage  that  is  fed  on  a  frame  /. 
The  hob  swivels  upon  the  carriage,  so  that 
the  tangent  to  its  pitch  spiral  can  be  set 
parallel  with  the  direction  of  the  feed,  and 
the  frame  swivels  so  that  the  tooth  can  be 
cut  at  any  angle  with  the  gear  spindle. 

As  the  blank  and  the  hob  are  revolving, 
the  latter  is  fed  into  the  former,  and  it  will 
cut  a  perfect  tooth  in  the  direction  that  the 
frame  is  set  at.  As  the  frame  can  be  set  in 
any  direction,  the  machine  will  cut  the  com- 
mon straight  tooth,  as  shown  by  Fig.  106. 
All  gears  cut  by  the  same  cutter  will  run 
together  interchangeably,  and  if  two  spiral 
gears  are  cut  at  the  same  angle  in  opposite 
directions  they  will  run  together  on  parallel 
shafts.  See  U.  S.  patent  number  405,030, 
June  llth,  1889. 


7.    IRREGUIvAR    AND    ELLIPTIC    GEARS. 


126.— NON-CIRCULAR  PITCH  LINES. 


The  consideration  of  pitch  lines  that  are 
not  circular,  and  of  the  teeth  that  are  fitted  for 
them,  is  an  interesting  but  not  particularly 
important  branch  of  odontics.  Such  pitch 


lines  are  largely  used  for  producing  variations 
of  speed  and  power,  but  have  no  other  prac- 
tical applications. 


127. — THE  IRREGULAR  PITCH  LINE. 

The  most  general  case  is  that  of  two  indefi- 
nite irregular  curves  rolling  together,  Fig. 
107,  the  only  condition  being  that  they 
shall  be  so  shaped  that  they  will  roll  together 
continuously. 

As  the  practical  importance  of  the  free 
pitch  line  is  very  small,  we  shall  not  ex- 
amine it  in  detail. 


Irregular  pitch  lines 

Fig.  107. 


128. — PITCH  LINES  ON  FIXED  CENTERS. 


When  we  attach  the  condition  that  the  two 
pitch  lines  shall  revolve  in  rolling  contact  on 
fixed  centers,  we  have  a  definite  problem  of 
more  interest  and  importance  than  that  of  the 
free  pitch  line. 

If,  as  in  Fig.  108,  we  have  a  pitch  line  A 
revolving  upon  a  fixed  center  a,  we  can  con- 
struct a  pitch  line  B  that  will  roll  with  it, 
and  revolve  on  the  given  fixed  center  b,  by 
the  following  process. 

From  any  pitch  point  0,  step  off  equal  arcs 
Oc,  cc,  cc ;  draw  circular  arcs  cd  from  the 
center  a;  draw  circular  arcs  dn  from  the 
center  b;  step  off  the  same  equal  arcs  Oe, 
ee,  ee,  then  Oeee  will  be  the  required  mat- 
ing pitch  line. 

These  curves  will  always  be  in  rolling  con- 
tact at  a  point  on  the  line  of  centers  ab,  the 
pitch  point  and  the  angle  of  the  curves  with 
the  line  of  centers  continually  changing. 

The  velocity  ratio  of  the  curves  will  be 


Fixed  centers 

Fig.  108. 

variable,  and  always  equal  to  the  inverse  pro- 
portion of  any  two  mating  radiants,  ac  and  be. 


68 


Multilobes. 


129. — CLOSED    PITCH    LINES. 

When  one  of  the  curves  of  Fig.  108  is  a 
closed  curve,  the  other  will  in  general  not 
be  closed,  but  by  trying  different  centers,  a 
curve  can  be  found  that  will  be  closed. 

If  the  closed  curve  a,,  Fig.  109,  is  taken, 
the  mating  curve  Al  will  be  closed  when  the 
center  is  chosen  at  a  certain  point  Bl,  that  can 
be  found  by  repeated  trials. 

The  mating  closed  curves  thus  constructed 
will  seldom  be  alike,  but  will  always  have 
points  of  similarity.  A  salient  point  q  on  one 
will  be  paired  with  a  reversed  point  or  notch 
on  the  other,  and  lobes  on  one  will  be  repre- 
sented by  depressions  on  the  other.  Half  a 
revolution  of  one  of  the  curves,  from  any 
position,  will  turn  the  other  through  half  a 
revolution. 


Set  of  Multilobes 

Fig.  109. 


1 30.  — M  ULTILOBES. 


If,  after  finding  the  center  Blt  Fig.  109,  for 
the  closed  mating  curve,  other  centers  are 
tried,  second,  third,  and  succeeding  centers, 
B,,  BZ,  #4,  will  be  found,  about  which  the 
mating  curves  will  also  be  closed. 

These  closed  curves,  called  multilobes,  will 
be  each  divided  into  like  lobes,  the  second 
curve,  or  bilobe,  into  two  lobes  ;  the  third,  or 
trilobe,  into  three  lobes,  and  so  on. 

If  the  center  is  placed  at  infinity,  the  rack 
lobe  A  oo  will  be  formed. 

If  the  center  be  taken  negatively,  on  the 
same  .side  as  the  original  center  blf  at  Z>3, 
#3,  64,  etc.,  negative  multilobes  a»,  a3,  »4, 
etc.,  will  be  formed  about  the  original  curve 
a,. 

All  these  multilobes,  positive  and  negative, 
will  roll  together  collectively  about  their  fixed 
centers,  in  rolling  contact  at  a  common  and 
shifting  pitch  point  0. 

Any  two,  of  the  same  sign,  will  roll  in  inter- 
nal contact,  and  any  two  of  opposite  signs 


Train  of  tnultilobcs 

Fig.  110. 

will  roll  in  external  contact,  so  that  they  can 
be  formed  in  train,  Fig.  110. 

When  it  so  happens,  as  it  does  with  the 
ellipse  revolving  on  its  focus,  or  the  logarith- 
mic spiral  revolving  on  its  pole,  is  taken,  that 
the  first  derived  pair  of  curves,  or  unilobes, 
are  exactly  alike,  all  the  multilobes  will  be 
alike ;  the  positive  trilobe  like  the  negative 
trilobe,  and  so  on,  so  that  any  two  curves  of 
such  a  set  will  work  together  in  either  inter- 
nal or  external  contact,  Fig.  111. 


Conic  Pitch  Lines. 


69 


131. — CONIC   SECTION  PITCH  LINES. 


If  two  like  conic  sections  are  mounted  upon 
their  foci,  they  will  roll  together. 

Their  free  foci  will  revolve  at  a  fixed  dis- 
tance from  each  other,  and  may  be  connected 
by  a  link.  The  line  of  the  free  foci  will  in- 
tersect the  line  of  the  fixed  foci  at  the  point 
of  contact  of  the  pitch  lines. 

Fig.  112  shows  a  pair  of  ellipses,  Fig.  113 
a  pair  of  parabolas,  and  Fig.  114  a  pair  of 
hyperbolas. 

The  elliptic  pitch  line  is  the  only  one 
known  that  will  revolve  with  its  equal,  and 
make  a  practical  and  complete  revolution. 


fixed 


Elliptic  tnultilobea 

Fig.  111. 


Ellipticpitch  lines 

Fig.  112. 


Parabolic  pitch  lines 

Fig.  113. 


Hyperbolic  pitch  Hues 


>       Of 

IIVEE3ITY, 


70 


Logarithmic  Pitch  Lines. 


132. — THE  LOGARITHMIC  SPIRAL. 


If  the  radiants  a,  b,  c,  d,  e,  Fig.  115,  make 
equal  angles  with  each  other,  and  each  one 
is  equal  to  the  adjacent  one  multiplied  by  a 
constant  number,  their  extremities  will  deter- 
mine a  logarithmic  spiral. 

If  the  first  radiant  a  is  given,  with  the  con- 
stant multiplier  ??,  the  second  radiant  will  be 
na,  the  third  will  be  n*  a,  the  fourth  will  be 
n9  a,  and  so  OD. 

If  the  first  and  last  radiants,  a  and  e,  are 
given,  and  there  are  p  equal  angles  between 
them,  the  constant  is 


so  that  it  is  a  simple  matter  to  construct  a 
logarithmic  spiral  to  connect  any  two  given 
radiants  at  any  given  angle  with  each  other. 

The  curve  possesses  the  singular  property 
that  all  tangents,  A  or  E,  make  the  same 
angle  with  the  radiants  at  their  points  of  con- 
tact. The  curves  are  always  inclined  to  the 
line  of  centers  at  the  constant  angle. 

The  curve  continually  approaches  the 
center  M,  or  "pole,"  making  an  infinite 
number  of  turns  about  it,  but  ntver  reach- 
ing it. 

It  also  has  the  entirely  useless  property  that 
the  pole  will  trace  an  involute  of  the  base 
circle  if  it  is  rolled  upon  the  pitch  circle  (75). 

It  possesses  the  property,  not  possessed  by 
any  other  curve,  that  it  will  roll  with  an 
equal  mate  on  fixed  centers  that  can  be  varied 
in  position.  The  curve  H  will  roll  with  the 
curve  C,  whether  its  pole  is  at  N,  or  at  8,  or 
at  F. 

Fig.  116  shows  a  pair  of  logarithmic  spi- 
rals in  internal  contact. 


Logarithmic  pitch  lines 

Fig.  115. 


Internal  logarithmic 
pitch  lines 

Fig.  1W. 


133. — COMPOSITE   PITCH   LINES. 


Instead  of  drawing  a  curve  at  random, 
and  finding  the  mate  to  run  with  it,  Fig. 
108,  the  complete  pitch  line  may  be  built 
up  of  a  number  of  curves,  of  which  the 
properties  are  known. 

Thus,  Fig.  117  shows  composite  gears, 
consisting  of  circular  parts  A  and  a,  and  an 
elliptic  trilobe  B,  working  with  an  elliptic 


bilobe  b.  Fig.  118  shows  a  combination  of  a 
pair  of  logarithmic  spiral  arcs  A  and  a,  a 
pair  of  elliptic  bilobal  arcs  B  and  b,  a  pair 
of  logarithmic  spiral  arcs  D  and  d,  and  a 
pair  of  elliptic  quadrilobal  arcs  E  and  e. 
An  endless  variety  of  combinations  can  be 
made  in  this  way. 
It  is  not  necessary  that  the  component 


Composite  Pitch  Lines. 


71 


curves  be  tangent,  if  they  succeed  each  other 
continuously.     Fig.  119  shows  a  pair  of  equal 
logarithmic  spirals  with  a  break  at  ab,  the 
action  at  b  commencing  just  as  it  ends  at  a. 
Care  should    be   taken  to  avoid    salient 


points,  breaks,  and  interruptions  of  the  con- 
tinuity of  the  curve,  for  there  must  be 
defective  tooth  action  at  such  points.  The 
cuives  should  run  smoothly  into  each  other 
with  gradual  changes  of  curvature. 


Composite  pitch  lines 
Fig. 


Composite  pitch  lines 

Fig. 


Broken  pitch  lines 

Fig.  119. 


134. — TEETH   OF    NON-CIRCULAR   PITCH   LINES. 


The  action  of  the  teeth  of  non-circular 
pitch  lines  does  not,  at  first  sight,  appear  to 
follow  the  laws  pertaining  to  circular  lines, 
but  there  is  really  very  little  difference. 

If  we  consider  the  two  pitch  lines  to  be 
free,  and  to  be  so  moved  while  they  roll 
together  that  the  pitch  point  0,  Fig.  107,  is 
fixed,  and  so  that  the  fixed  line  c  C  is  always 
at  right  angles  to  both  curves  at  their  com- 
mon point  0,  the  laws  of  the  tooth  action  will 
be  almost  precisely  the  same  as  laid  down 
for  the  circular  pitch  line.  Fig.  107  may  be 
easily  applied  to  (24)  as  illustrated  by  Fig.  15. 

When  the  centers  are  fixed,  the  same  tooth 


action  takes  place,  but  the  line  of  action  and 
the  pitch  point  continually  change  their 
positions. 

The  teeth  of  non-circular  pitch  lines  can 
therefore  be  formed  either  by  conjugating  a 
given  odontoid,  as  in  (24),  or  by  the  roiled 
curve  theory  of  (32). 

By  all  means  the  most  practicable  method, 
when  the  circumstances  will  permit,  is  to 
make  up  the  curve  by  joining  approximating 
circular  arcs,  and  to  provide  each  circular 
arc  with  teeth  in  the  ordinary  way.  See  this 
process  as  applied  to  the  elliptic  pitch  line  at 
Figs.  129 -and  130. 


135. — TEETH   AT    SALIENT  POINTS   AND   BREAKS. 


When  there  is  a  salient  point,  or  other  inter- 
ruption of  the  continuity  of  the  action,  as  at  q, 
Fig.  109,  or  at  Mm,  Fig.  118,  there  must  be 


an  interruption  in  the  arrangement  of  the 
normals  of  any  tooth  curve,  and  a  consequent 
failure  of  the  tooth  action. 


1-2 


Elliptic  Pitch  Lines. 


Fig.  120  shows  a  cycloidal  tooth  curve  Jf, 
at  a  corner  or  salient  point  S,  between  two 
circular  pitch  arcs.  There  is  a  circular  arc 
A  on  the  odontoid  made  while  the  describing 
circle  is  turning  about  the  point  8,  and  that 
arc  can  have  no  continuous  tooth  action. 
Therefore  the  tooth  action  will  fail,  unless 
the  next  tooth  curve  .2V  springs  from  the 
salient  point. 

If  a  tooth  springs  from  the  salient  point, 
the  tooth  action  will  be  correct,  but  mechani- 
cally imperfect,  as  the  arc  of  action  of  two 
teeth  cannot  lap  over  each  other  to  allow  for 
practical  defects.  And  then,  as  two  tooth 
curves  cannot  spring  from  the  same  pitch 
point  in  opposite  directions,  such  gears  can 
run  in  but  one  direction,  and  are  not  reversi- 
ble. 

When  there  is  a  break,  as  at  ab,  Fig.  119, 
the  teeth  must  be  so  cut  off  that  they  will 


The  salient  point 

Fig.  12O. 


separate  at  a  just  as  they  engage  at  b,  for  there 
is  a  sudden  change  in  the  velocity  ratio. 
Such  combinations  are  practicable,  but  in 
every  way  undesirable. 


136. — THE   ELLIPTIC    GEAR. 


The  principal,  and  almost  the  only  use  of 
the  irregular  gear,  is  to  produce  a  variation 
of  speed  between  certain  given  limits,  with- 
out conditions  as  to  the  variations  of  speed 
and  details  of  the  motion  between  the  limits. 
When  that  is  the  only  object,  the  elliptic 
pitch  line  is  the  only  one  that  is  required,  and 
it  is  chosen  because  it  is  the  only  known  con- 
tinuous closed  curve  that  will  work  in  roll- 
ing'contact  with  an  equal  mate,  and  because 
it  is,  next  to  the  circle,  the  simplest  known 
curve.  Of  the  elliptic  multilobes,  the  uni- 
lobe,  or  simple  ellipse,  revolving  on  one  of 
its  foci  as  a  center,  is  the  only  one  used  to 
any  appreciable  extent,  and  therefore  is  the 
only  one  that  requires  examination  in  detail. 

The  use  of  the  elliptic  gear  is  practically 
confined  to  producing  a  simple  variation  of 
speed  between  known  limits,  and  to  produc- 
ing a  "quick  return  motion"  for  planers, 
shapers,  slotters,  and  similar  cutting  tools,  as 
well  as  for  pumps,  shears,  punches,  shingle 
machines,  and  others  where  the  work  is  done 
mostly  during  one-half  of  the  stroke  of  a 
reciprocating  piece.  The  work  of  a  planer 


tool  or  of  the  plunger  of  a  single  acting 
pump,  is  all  done  during  the  motion  of  the 
tool  or  of  the  plunger  in  one  direction,  and 
the  only  object  on  the  return  is  to  get  the 
piece  ready  for  the  next  useful  operation  in 
the  quickest  possible  time. 

For  an  example,  the  bobbin  of  a  spinning 
machine  is  to  be  wound  in  a  conical  form, 
the  thread  being  fed  to  it  through  a  moving 
guide,  and  the  necessary  variable  motion  of 
the  guide,  fast  at  the  point  of  the  cone,  and 
slow  at  its  base,  is  best  given  to  it  by  a  pair 
of  elliptic  gears.  For  another  example,  the 
motion  of  the  platen  of  a  printing  press 
should  be  rapid  when  the  press  is  open,  and 
slow  and  powerful  when  the  impression  is 
being  taken,  and  the  object  can  be  reached 
best  by  a  pair  of  elliptic  gears  operating  the 
platen. 

The  practical  uses  of  the  elliptic  gear  are 
endless,  and  it  would  be  in  greater  use  and 
favor,  if  it  were  not  for  the  fact  that  its  pro- 
duction, by  the  means  ordinarily  in  use  for 
that  purpose,  is  as  difficult  and  costly  as  the 
resulting  gear  is  unsatisfactory. 


Elliptic  Pitch  Lines. 


73 


137. — THE 

To  thoroughly  understand  the  construc- 
tion and  operation  of  the  ellipse,  it  is  neces- 
sary to  learn  but  a  few  of  its  many  proper- 
ties. 

The  mechanical  definition  of  the  ellipse  is 
that  it  is  one  of  the  "  conic  sections."  If  the 
cone,  Fig.  121,  is  cut  by  a  plane  G  at  right 
angles  with  its  axis,  the  outline  of  the  section 
will  be  a  circle;  if  the  plane  Ecuts  the  cone  at 
an  angle,  the  section  will  be  an  ellipse;  if  the 
plane  P  is  parallel  with  the  side  of  the  cone, 
the  section  is  a  parabola,  and  if  the  plane  H 
is  at  such  an  angle  that  it  cuts  both  nappes  of 
the  cone,  the  section  is  a  hyperbola.  All 
these  curves  will  roll  together  when  fixed  on 
centers  at  certain  points  called  foci,  but  the 
ellipse,  and  its  special  case,  the  circle,  are  the 
only  ones  that  are  capable  of  continuous  mo- 
tion. 

In  the  ellipse,  Fig.  122,  the  point  C  is  the 
center,  the  longest  diameter,  AA ',  is  the 
major  axis,  the  shortest  diameter,  BB' ,  is  the 
minor  axis;  A  and  A  are  the  major  apices, 
and  B  and  B'  are  the  minor  apices. 

If  an  arc  be  drawn  from  the  minor  apex, 
with  a  radius  equal  to  the  major  semi-axis,  it 
will  cut  the  major  axis  at  points  F  and  F, 
called  the  foci,  and  one  focus  must  be  chosen 
as  the  center,  about  which  the  curve  is  to  re- 
volve if  used  as  the  pitch  line  of  a  gear. 

It  is  a  property  of  the  curve  that  the  sum 
of  the  distances,  PF  and  PF',  from  any 
point  to  the  foci  is  equal  to  the  major  axis, 
AA',  and  this  feature  is  used  as  a  means  of 
constructing  the  curve  by  points.  Draw  any 
arc  at  random  from  one  focus  with  radius 
FP.  Draw  an  arc  from  the  other  focus  with 
a  radius  equal  to  A  A — FP,  and  it  will  cut 
the  first  arc  at  a  point  of  the  ellipse.  When 
the  point  P  is  near  either  major  apex,  the 
arcs  intersect  at  such  a  sharp  angle  that  the 
method  is  nearly  useless. 

Another,  and  much  the  best  known  method 
for  constructing  the  ellipse  by  points, is  to  draw 
any  radial  line  L,  and  also  circular  arcs  W 
and  V,  from  the  center  through  the  apices. 
From  the  intersections,  w  and  v,  of  the  radial 
line  and  the  circles,  draw  lines  parallel  to  the 
axes,  and  they  will  intersect,  always  at  right 
angles,  at  a  point  u  on  the  curve.  This 


ELLIPSE. 


Conic  sections 

Fiy.  121. 


The  ellipse 

Fig.  122. 


method  is  very  accurate,  and  has  no  failing 
position. 

Another  valuable  property  of  the  ellipse  is 
that  if  the  line  pab  be  so  drawn  that  the  dis- 
tance pa  is  equal  to  BC,  and  pb  to  AC,  the 
point  p  will  be  upon  the  curve  if  the  points  a 
and  b  are  upon  the  axes. 

The  curvature  of  the  ellipse  is  an  important 
feature  in  connection  with  its  use  as  a  gear 
pitch  line.  It  is  sharpest  at  the  major  axis 
A,  and  flattest  at  the  minor  apex  B,  else- 
where varying  between  the  two  limits. 

The  radius  of  curvature  at  either  apex, 
that  is,  the  radius  of  tne  circle  that  most 
nearly  coincides  with  the  curve,  is  found  by 
drawing  the  lines  Bk  and  Ak  at  right  angles 


74 


Elliptic  Pitch  Lines. 


with  the  chord  AB.  The  distance  Ch  is  the 
radius  of  curvature  at  the  major  apex  A,  and 
the  distance  Ck  is  the  radius  at  the  minor 
apex  B. 


The  normal  PN  to  the  curve  at  any  point 
P  bisects  the  angle  FPF'  between  the  focal 
lines,  and  the  tangent  PT  is  at  right  angles 
to  the  normal. 


138.— ELLEPTOGRAPHS. 


There  are  a  multitude  of  elliptographs,  or 
instruments  for  drawing  the  ellipse,  but  only 
two  of  them  are  of  practical  application  in 
this  connection. 

The  simplest  known  elliptograph  consists 
of  a  couple  of  pins,  a  thread,  a  pencil,  and  a 
stock  of  patience.  The  pins  are  inserted  at 
the  foci,  as  in  Fig.  123,  and  the  curve  is 
drawn  by  moving  the  pencil  with  a  uniform 
strain  against  the  string.  After  a  number  of 
trials,  depending  in  number  on  the  skill  of  the 
draftsman,  the  curve  may  be  induced  to  pass 
through  the  desired  points.  The  best  result 
will  be  obtained  by  the  use  of  a  well  waxed 
thread  running  in  a  groove  near  the  point 
«of  a  hard  pencil.  The  pencil  should  be  long, 
.and  held  by  the  end,  so  that  the  strain  on  the 
string  will  be  uniform,  for  the  elasticity  of 
the  string  is  the  greatest  source  of 
error.  This  "  gardener's  ellipse  " 
will  generally  be  accurate  enough 
for  a  tulip  patch,  but  should  not 
be  relied  upon  for  mechanical  pur- 
poses, unless  one  or  more  points 
between  the  apices  are  tested  and 
found  to  be  correct.  If  the  two 
pins  and  the  pencil  are  circular, 
and  of  the  same  diameter,  ~the  ac- 
curacy of  the  ellipse  is  independent 
'  of  their  diameter. 

The  best  elliptograph  is  the 
"trammel,"  Fig.  124,  which  takes 
a  variety  of  shapes,  but  which  in 
its  simplest  condition  consists  of 
a  cross,  with  two  grooves  at  right 
angles,  and  a  bar  D  with  two  pins 
a  and  b,  and  a  tracing  point  P 
placed  in  line.  The  distance  Pb 
being  set  to  the  major  semi-axis,  and 
the  distance  Pa  to  the  minor  semi- 
axis,  the  point  P  will  trace  the  ellipse  if  the 
pins  are  confined  to  move  in  the  grooves. 
±lf  carefully  made,  the  instrument  works 


with  great  precision,  is  easily  handled  and 
set,  and,  if  the  curve  drawn  is  not  very 
flat,  it  may  be  inked.  The  cheap  wooden 


Fig.  123. 


Gardener's  ellipse. 


The  trammel 

Fig.  124=. 


trammel  should  not  be  tolerated,  for  the 
string  and  two  pins  cost  less  and  are  more 
reliable. 


Elliptic  Pitch  Lines. 


75 


139. — APPROXIMATE   CIRCULAR  ARCS. 


If  a  well-made  trammel  is  not  at  hand,  the 
best  plan  is  to  draw  the  ellipse  with  a  string, 
through  several  constructed  points,  and  then 
to  ink  it  by  finding  centers  for  approximate 
arcs,  as  shown  by  Fig.  125.  An  arc  from  a 
center  m  on  the  major  axis,  will  coincide 
very  well  with  the  curve  near  the  major 
apex,  a  similar  arc  n  from  a  center  on  the 
minor  axis  will  serve  near  the  minor  apex, 
and  a  third  center  q  can  be  found  for  an  arc 
to  join  the  first  two.  More  than  three  cen- 
ters will  seldom  be  required,  and  when  the 
ellipse  is  not  very  flat  the  two  centers  on  the 
axes  will  be  sufficient. 


.  125. 


The  elliptic  involute 


140. — FOUR  CENTER  ELLIPSE. 


When  the  ratio  of  the  axes  is  not  less  than 
eight  to  ten,  as  is  generally  the  case,  a  prac- 
tically perfect  ellipse  may  be  drawn  from 
four  centers  by  the  following  method. 

Draw  the  line  CL,  Fig.  126,  parallel  to 
A'B,  and  construct  the  point  u  on  the  ellipse 
by  the  method  of  (137).  Find  a  point  a  on 
the  major  axis,  from  which  an  arc  from  A 
will  pass  through  u,  and  it  will  be  the  major 
center.  It  may  be  found  by  trial,  or  by 
drawing  um  at  right  angles  to  uA,  and 
bisecting  Am  in  a. 

Through  a  draw  ac  at  right  angles  to  AB, 
and  its  intersection  with  the  minor  axis  will 
be  the  minor  center  b.  Lay  off  Co,'  and  CV 
equal  to  Ca  and  Cb,  and  draw  be',  b'c",  and 
b'c'". 

From  the  centers  a  draw  the  arcs  cAc'", 
and  e'A'c",  and  from  the  centers  b  draw  the 
arcs  cBc'  and  c"B'c'". 


Fig.  126. 


al 


JS' 

Four  center  method 


Lines  that  are  parallel  to  the  pitch  line, 
such  as  the  addendum,  root,  clearance,  and 
base  lines,  are  to.  be  drawn  from  the  same 
centers. 


141. — ROLLING   ELLIPSES. 


When  two  equal  ellipses,  Fig.  127,  are 
arranged  to  revolve  on  their  foci  as  centers, 
with  a  center  distance  equal  to  the  major 
axis,  they  will  roll  together  perfectly,  and 
be  fitted  to  act  as  the  pitch  lines  of  gear 
wheels. 


of  the  arrow  d,  it  will  drive  the  follower 
F  by  direct  contact  of  the  pitch  ellipses,  but 
when  turning  in  the  other  direction  with 
respect  to  the  follower,  as  it  must  during 
half  of  its  revolution,  it  has  no  direct 
driving  action,  and  the  follower  must  be 


When  the  driver  D  turns  in  the  direction  !  kept  in  contact  by  some  other  force. 


Spacing  tJic  Ellipse. 


As  the  two  ellipses  roll  together,  the  free 
foci  F%  and  F±  will  always  move  at  a  con- 
stant distance  apart,  equal  to  the  distance 
between  the  fixed  foci,  and  therefore  they 
may  be  connected  by  the  link  L. 

The  center  line  of  the  link  will  always 
cross  the  fixed  center  line  at  the  point  of  con-  j 
tact  of  the  ellipses,  and  the  tangent  T  at  that  | 
point  will  pass  through  the  intersection  of  the 
axes. 


142. — SPACING  THE  ELLIPSE. 

As  the  ellipses  roll  together  it  is  essential  |  If  the  ellipse  is  drawn  by  means  of  the 
that  the  axes  come  in  line,  and  therefore,  if  trammel,  Fig.  124,  it  can  be  accurately  spaced 
the  teeth  of  one  gear  are  fixed  at  random,  by  means  of  a  graduated  index  circle  /,  hav- 
those  of  the  other  must  be  fixed  to  corre-  j  ing. a  diameter  equal  to  the  sum  of  the  diam- 
spond.  If  this  requirement  is  satisfied,  it  j  eters  of  the  ellipse,  for  then  the  center  line  of 


makes  no 
placed. 


difference  where    the    teeth    are 


It  is,  however,  very  desirable  that  the  two 


the  bar  will  pass  over  an  arc  on  the  ellipse 
that  at  the  apices  is  exactly  equal  to  half  the 
arc  passed  over  at  the  same  time  on  the  circle, 


gears  shall  be  exactly  alike,  so  that  they 
can  be  cut  at  one  operation  while  mounted 
together  on  an  arbor  through  their  focus 
holes,  and  to  do  this,  it  is  necessary  to  start 
the  teeth  at  different  points,  according  to 
whether  their  number  is  odd  or  even. 

If  the  number  of  teeth  is  even,  one  tooth 
must  spring  from  the  major  axis,  as  shown 
by  Fig.  128. 

If  the  number  of  teeth  is  odd,  the  major 
axis  must  bisect  a  tooth  and  a  space,  as  shown 
by  Fig.  129.  In  this  case,  if  one  of  the  1  and  that  is  elsewhere  very  nearly  in  the  same 


Fig.  128. 


gears  can  be  turned  over,  or.if  its  other  focus 
hole  can  be  used  as  a  center,  it  may  have  a 
tooth  springing  from  the  major  axis. 

The  simplest  method  of  spacing  the  ellipse 
is  to  step  about  it  with  the  dividers.  If  the 
curve  is  flat,  the  dividers  should  be  set  to  less 
than  a  whole  tooth,  for  equal  chords  will  not 
measure  equal  arcs  of  the  curve. 

But  this  stepping  method,  although  it  is 
sufficient  and  convenient  for  drafting  pur- 
poses, is  wholly  unfit  for  mechanical  pur- 
poses, and  therefore  we  must  have  a  method 
that  is  not  dependent  on  personal  skill. 


proportion. 

This  method  is  not  mathematically  exact, 
but  its  accuracy  is  very  far  within  the  re- 
quirements of  practice.  The  space  on  the 
quarter,  at  Q,  will  be  greater  than  anywhere 
else,  but  the  maximum  error  will  in  general 
be  very  minute. 

For  an  example,  take  an  extreme  practical 
case,  a  gear  with  axes  eight  and  ten  inches 
long,  and  with  seventy-two  teeth  The  max- 
imum error,  the  difference  between  the  long- 
est and  shortest  tooth  arcs,  will  be  not  over 
one  five-hundredth  of  an  inch.  In  the  mor< 


Involute  Elliptic    Gears. 


77 


common  practical  case  of  a  gear  of  nine  and 
ten  inches  axes,  and  seventy-two  teeth,  the 
maximum  error  is  about  one  two-thousandth 
of  an  inch.  In  both  these  cases,  the  differ- 
ence between  the  tooth  arc  at  the  major  apex 
and  that  at  the  minor  apex  is  too  small  to  be 


readily  calculated,  but  will  be  about  one 
twenty-thousandth  of  an  inch.  In  all  cases 
likely  to  be  met  with  in  practice,  the  inevit- 
able mechanical  errors  are  greater  than  the 
theoretical  errors  of  the  method,  and  it  is 
serviceable  on  ellipses  as  flat  as  three  to  one. 


Involute  elliptic  teeth. 

Fig.  129. 

143. — INVOLUTE  ELLIPTIC  TEETH. 


As  in  the  case  of  the  circular  gear,  the 
best  form  of  tooth  for  the  elliptic  gear  is  the 
involute,  and  for  the  same  reasons. 

The  base  line  of  the  involute  tooth  is  any 
ellipse  BE,  Fig.  125,  which  is  drawn  from 
the  same  foci  as  the  pitch  ellipse  ;  the  limit 
point  i  is  the  point  of  tangency  of  a  tangent 
from  the  pitch  point  0,  and  the  addendum 
line  a  I  of  the  mating  gear  must  not  pass 
beyond  that  point.  The  method  of  laying 
out  the  tooth  and  drafting  it  is  so  exactly 
like  the  process  for  the  circular  gear  that 
the  description  need  not  be  repeated. 

The  centers  of  involute  elliptic  gears  can 
be  adjusted  without  affecting  the  perfection 
of  the  motion  transmitted,  but,  as  the  focal 


distance  remains  fixed,  the  ratio  of  the  axes 
will  be  altered. 

The  work  of  drawing  the  teeth  can  be 
much  abbreviated  by  the  process  illustrated 
by  Fig.  129.  Find  the  centers  for  approxi- 
mate circular  arcs,  preferably  by  the  method 
of  (140),  and  then  consider  the  gear  as  made 
up  of  four  circular  toothed  segments.  It  is 
then  necessary  to  construct  but  two  tooth 
curves,  one  for  the  major  and  one  for  the 
minor  segment,  and  the  flanks  will  be  radii 
of  the  circular  arcs. 

The  line  of  action,  la,  Fig.  125,  is  not  a 
straight  line,  and  it  is  not  the  same  for  all 
the  teeth.  It  is  not  fixed  when  the  pitch 
point  0  and  the  line  of  centers  is  fixed  (134). 


Cycloidal  Elliptic    Gears. 


Cycloldal  elliptic  teeth. 

Fig.  130. 

144.— CYCLOIDAL  ELLIPTIC  TEETH. 


The  cycloidal  tooth  is  drawn,  exactly  as 
upon  a  circular  pitch  line,  by  a  tracing  point 
in  a  circle  that  is  rolled  on  both  sides  of  the 
pitch  line.  The  line  of  action  is  not  a  circle, 
and  it  is  not  the  same  curve  for  all  the  teeth. 

That  the  flanks  shall  not  be  under-curved, 
the  diameter  of  the  rolling  circle  should  not 
be  greater  than  the  radius  of  curvature  at 


the  tooth  being  drawn,  and  when,  as  usual, 
the  same  roller  is  used  for  all  the  teeth,  its 
diameter  should  not  be  greater  than  the 
radius  of  curvature  at  the  major  apex,  the 
distance  Ch  of  Fig.  122. 

Fig.  130  shows  a  cycloidal  gear  drawn  as 
four  circular  segments,  by  the  methods  of 
(140)  and  (83). 


145. — IRREGULAR   TEETH. 


It  is  most  convenient  to  draw  all  the  teeth 
alike,  with  the  same  rolling  circle,  or  from 
the  same  base  line,  and  also  to  uniformly 
space  the  pitch  line,  but  such  uniformity  is 
not  essential. 


The  only  requirement  is  that  each  tooth 
curve  shall  be  conjugate  to  the  tooth  curve 
that  it  works  with,  and  if  that  condition  is 
satisfied  the  teeth  may  be  of  all  sorts  and 
sizes. 


146. — FAILURE   IN   THE   TOOTH  ACTION. 


When  the  major  axes  are  in  line  the  action 
of  the  teeth  on  each  other  is  nearly  direct, 
but  when  the  minor  axes  are  in  line  the  action 


is  more  oblique,  as  shown  by  Fig.  127.  The 
teeth  tend  to  jam  together  when  the  driver 
is  pushing  the  follower,  and  to  pull  apart 


The  Link. 


79 


when  the  follower  is  being  pulled,  and  when 
the  ellipse  is  very  flat  this  tendency  is  so 
great  that  the  teeth  fail  to  act  serviceably. 

At  first  glance  it  might  appear  that  this 
difficulty  in  the  tooth  action  of  very  eccentric 
gears  might  be  overcome  by  making  the  teeth 
radial  to  the  focus,  as  shown  by  Fig.  131, 
but  examination  will  show  that  but  little  can 
be  gained  in  that  way. 

The  teeth  on  the  gear  G  were  obtained  by 
the  method  of  (28)  from  the  assumed  tooth 
on  the  gear  c,  and  the  effect  of  the  defective 
shape  of  one  side  of  the  assumed  tooth  was  to 
cut  away  the  conjugate  curve  of  the  derived 
tooth. 

Such  teeth  would  not  work  as  well  as  the 
ordinary  form,  and  their  construction  would 
be  very  difficult. 


Uttrfial  teeth 

Fig.  131. 


147. — THE   LINK. 


When  the  teeth  of  the  elliptic  gear  fail  to 
properly  engage,  on  account  of  the  obliquity 
of  the  action,  the  difficulty  can  be  entirely 
overcome  by  connecting  the  free  foci  by  a 
link  (141),  as  shown  by  Fig.  127. 

This  link  works  to  the  best  advantage 
when  the  teeth  are  working  at  the  worst,  and 
when  it  fails  to  act,  as  it  passes  the  centers, 
the  teeth  are  working  at  their  best.  There- 


fore gears  that  are  connected  by  a  link  need 
teeth  only  at  the  major  apices. 

When  the  tooth  action  is  imperfect  by  rea- 
son of  its  obliquity,  and  the  link  is  not  avail- 
able or  desirable,  the  difficulty  can  be  over- 
come by  using  three  or  more  gears  in  a  train, 
as  shown  by  Fig.  137,  for  then  the  same  re- 
sult can  be  obtained  by  the  use  of  gears  that 
are  much  more  nearly  circular. 


148. — VARIABLE   SPEED   AND   P.OWER. 


If  the  shaft  c,  Fig.  132,  turns  uniformly, 
the  slowest  speed  of  the  shaft  C  will  occur 
when  the  gears  are  in  the  position  of  the 
figure,  and  the  proportion  between  the  two 
speeds  will  be  the  proportion  between  the 
distances  cO  and  CO.  The  greatest  speed 
of  the  driven  shaft  will  occur  when  the  shafts 
have  turned  through  a  half  revolution  from 
the  position  of  the  figure,  and  the  relative 
speed  will  be  the  same,  reversed. 


Fig.  132. 


The  ratio  of  speed,  the  ratio  of  the  greatest   variation  of  the  axes  to  produce  a  decided 


speed  to  the  slowest  speed,  is  the  square  of 
the  ratio  between  the  speed  of  the  driving 
shaft  and  the  greatest  or  the  least  speed  of  the 
driven  shaft,  so  that  it  requires  but  a  slight 


variation  of  the  speed. 

The  following  table  will  give  the  propor- 
tion of  minor  to  major  axes  that  will  give 
any  desired  ratio  of  speeds. 


80 


Elliptic  Quick  Return  Motion. 


Ratio  of  Speeds. 

2  

3 

4 

5 

6 

7... 


Ratio  of  Axes. 

985 

.    .962 


.907 
.892 
.878 
.868 
.854 
.844 
.8M 
.824 
.817 
.807 
.800 


The  power  is  always  inversely  proportional 
to  the  speed.  If  the  variable  shaft  is  running 
twice  as  fast  as  the  uniform  shaft,  it  will  ex- 
ert but  one-half  the  force. 


When  the  gears  are  arranged  in  a  train,  as 
in  Fig.  137,  the  speed  ratio  for  the  second, 
third,  and  following  gears  will  be  in  the  pro- 
portion of  the  first,  second,  third  and  follow- 
ing powers  of  the  first  ratio. 

Thus,  the  ratio  for  a  pair  of  gears  with 
axes  in  the  proportion  of  .952  to  1  being  4 
for  the  second  gear,  will  be  16  for  the  third 
gear,  64  for  the  fourth  gear,  and  so  on. 

The  use  of  gears  of  troublesome  eccentric- 
ity can  be  avoided  by  this  means.  A  train 
of  three  gears  of  .952  axes,  Fig.  137,  is 
equivalent  to  a  single  pair  of  very  flat  gears 
with  .800  axes,  Fig.  138,  and,  in  general, 
three  gears  that  are  nearly  circular  are  equiva- 
lent to  a  single  very  flat  pair. 


149.— ELLIPTIC   QUICK   RETURN   MOTION. 


If  the  gears  are  arranged  with  respect  to 
the  piece  to  be  reciprocated,  in  the  manner 
shown  by  Fig.  133,  the  time  of  the  cutting 
stroke  will  be  to  the  time  of  the  return  stroke, 
as  the  angle  PEK  is  to  the  angle  PEF, 
where  J^and  F  are  the  foci  of  the  ellipse. 

The  following  table  will  show  the  ratio  of 
axes  that  must  be  adopted  to  produce  a  re- 
quired ratio  of  stroke  to  return. 

Quick  Return.  Ratio  of  Axes. 

J2tol 964 

3  tol 910 

4tol 861 

5  tot 817 

6  tol 778 

To  determine  the  ellipse  that  will  give  a 
required  quick  return,  we  lay  off  the  angles 
PEK  and  PEF  in  the  given  proportion, 
and  then  find  by  trial  a  point  P  such  that  the 
length  PE  plus  the  length  of  the  perpendicu- 
lar PF  is  equal  to  the  known  center  distance 
Ee.  F  will  be  the  other  focus  of  the  re- 
quired ellipse. 

When  the  driving  gear  has  turned  through 
the  angle  P'EF,  from  the  position  of  the 
figure  at  the  middle  of  the  return,  the  varia- 
ble gear  will  have  turned  through  the  angle 
P"eO  =  P'FO,  and  we  can  study  the  action 
of  the  tool  by  drawing  equi-distant  radii 
about  E,  and  finding  the  corresponding  radii 
about  F. 


Quick  return 

Fig.  133. 


Fig.  134  shows  the  arrangement  of  the 
radii  (P'F  =  F'e  of  Fig.  133)  in  the  case  of  a 
four  to  one  quick  return,  and  it  is  seen,  by 
the  parallel  lines,  that  the  motion  of  the  tool 
is  very  uniform,  coming  quickly  to  its  maxi- 
mum speed,  and  holding  a  quite  uniform 
speed  until  near  the  end  of  the  stroke.  Fig. 
135  shows  that  the  same  motion  derived  from 
a  simple  crank  is  not  as  uniform. 

When  the  gears  are  arranged  in  a  train, 
Fig.  137,  the  quick  return  ratios  can  be  de- 
termined by  the  construction  shown  by  Fig. 
136.  Draw  Fc  at  right  angles  to  A  A',  and 
draw  cEd  through  the  other  focus.  The 
quick  return  ratio  of  the  second  gear  will  be 
the  ratio  of  the  angles  aa  and  ba.  Draw 
dFe,  and  the  ratio  for  the  third  gear  will  be 


Elliptic   Trains. 


81 


Quick  return  crank 

.  134. 


Ordinary  crunk 

Fig.  135. 


that  of  the  angles  «8  and  &„.  Draw  eEf, 
and  «4  and  Z>4  will  give  the  ratio  for  the 
fourth  gear.  And  so  on,  in  the  same  man- 
ner, as  far  as  desired,  the  ratio  being  greatly 
increased  by  each  gear  that  is  added  to  the 
train. 

If  carefully  performed,  the  graphical  pro- 
cess is  quite  accurate.  The  case  of  axes  in 
the  proportion  of  .98  to  1  gave  a  quick  re- 
turn of  1.6  for  the  second  gear,  and  2.8  for 
the  third  gear,  while  their  true  computed 
values  are  1.66  and  2.74. 

The  chart  will  solve  quick  return 
train  questions  involving  gears  not 
flatter  than  .80,  as  accurately  as  need 
be.     For  example,  the  ratio  of  axes 
of  .95  will  give  a  quick  return  of 
2.25  for  the  second  gear,  4.85  for 
the  third  gear,   9.80  for  the  fourth    ' 
gear,  and  19.70  for  the  fifth  gear.     Again, 
the  proportion  of  axes  to  give  a  quick  return 
of  5  for  the  third  gear  is  .948. 


Quick  return  train 

Fig.  130. 


Elliptic  train      Fig.  137 » 


Fig.  138. 


Elliptic    Gear    Cutting  Machine. 


Elliptic  Quick  Return  Chart 


,81     .82    .83     .84     .85 


.87,    *88    ..89    .90     .91    .92     .S 
Proportion  of  Axes 


.94     .95    .96     .97    .98     .99    1.00 


150. — THE   ELLIPTIC   GEAR    CUTTING 
MACHINE. 

The  conditions  of  the  described 
-operation  of  drawing  the  ellipse  by 
means  of  the  trammel  (138)  may  be 
reversed,  the  bar  being  held  still 
while  the  paper  and  the  cross  are 
revolved,  and  it  is  evident  that  the 
xesult  will  be  the  same  ellipse  on  the 
paper  as  if  the  bar  is  revolved  as 
described. 

By  thus  reversing  the  process  of 
describing  the  ellipse,  and  by  adopt- 
ing the  improved  spacing  device  of 
(142),  we  can  construct  a  machine 
for  accurately  cutting  the  teeth  in 
an  elliptic  gear,  the  main  features 
of  which,  omitting  various  unessen- 
tial details,  are  shown  by  Figs.  139 
and  140. 

The  blank  to  be  cut  is  fastened 
upon  a  trammel  stand,  which  cor- 
responds to  the  paper  in  the  graphi- 
cal process,  and  revolves  upon  the 
fixed  base.  The  adjustable  trammel 
pins  a  and  b  are  fixed  in  a  slot  in 
the  bed,  and  they  fit  and  slide  in  the 
slots  M  and  N  in  the  under  surface 
of  the  stand.  The  cutter  which 
corresponds  to  the  tracing  point  is 
fixed  with  the  pitch  center  of  its 


Plan 
Fig.  139. 


Cutter 


Elevation 
Fig.  14O. 


Elliptic  Bevel   Gear. 


83 


tooth  curve  directly  over  the  point  P  in  the 
line  of  the  pins.  The  index  plate  has  a 
diameter  equal  to  the  sum  of  the  axes  of  the 
ellipse,  and  it  is  held  by  an  index  pin  p, 
which  slides  in  the  slot,  and  is  always'in  the 
line  of  the  pins. 

Thus  arranged,  the  machine  will  always 
cut  its  tooth  in  the  true  ellipse,  and  the  teeth 
will  be  accurately  spaced. 

The  direction  of  the  tooth  will  be  sub- 
stantially at  right  angles  to  the  pitch  line, 
and  a  simple  arrangement  can  be  applied  to 
make  it  exactly  so.  An  index  plate  of  a 
fixed  diameter  may  be  used  for  all  sizes  of 


gears,  if  the  index  pin  is  carried  by  an  arm 
which  swings  about  the  center  of  the  gear, 
and  has  an  adjustable  pin  that  slides  in  the 
slot. 

The  tops  of  the  teeth  are  trued  by  a  cutter 
having  a  square  .edge,  and  the  line  of  the 
tops  will  be  substantially  parallel  to  the  pitch 
line. 

The  blank  is  held  by  an  arbor  through  its 
focus  hole,  and  the  arbor  is  held  by  a  slide, 
which  slides  in  a  chuck  upon  the  stand,  so 
that  the  focus  can  be  accurately  set  in  the 
major  axis  at  the  proper  distance  from  the 
center. 


151. — CHOICE   OF    CUTTERS. 


Theoretically,  the  teeth  are  of  different 
shapes,  as  they  are  in  different  positions  upon 
the  ellipse,  and,  therefore,  each  space  should 
be  cut  with  a  cutter  that  is  shaped  for  that 
particular  space.  But  as  this  is  impracticable, 
it  is  necessary  to  choose  the  cutter  that  will 
serve  the  best  on  the  average. 

Strictly,  the  cutter  should  be  the  one  that 
is  fitted  to  cut  a  spur  gear  having  a  pitch 
radius  equal  to  the  radius  of  curvature  of 


the  ellipse  at  the  major  apex,  but  as  that 
cutter  will  be  much  too  rounding  for  the 
minor  apex,  it  is  better  to  choose  the  one 
that  is  fitted  for  the  medium  radius  of  cur- 
vature. 

The  two  radii  of  curvature  are  the  dis- 
tances Oh  and  Ck,  Fig.  122,  and  the  cutter 
should  be  chosen  for  the  radius  half  way 
between  the  two,  approximately  half  the 
sum  of  the  two. 


152. — THE  ELLIPTIC  BEVEL  GEAR. 


An  ellipse  may  be  drawn  on  the  surface  of 
a  sphere  by  means  of  a  string  and  two  pins, 
according  to  the  method  of  (138),  and  a 
pair  of  such  spherical  ellipses  will  roll  on 
each  other  while  fixed  on  their  foci,  their 
free  foci  moving  at  a  constant  distance 
apart. 

Therefore  we  can  have  elliptic  bevel 
gears  that  are  very  similar  to  elliptic  spur 
gears,  as  shown  by  Fig.  141.  The  two  gears 
revolve  on  radial  shafts  through  their  foci, 
and  the  link  connects  radial  shafts  through 
the  free  foci.  The  velocity  ratio  is  the  ratio 
of  the  perpendiculars  a  b  and  a  c.  The 
elliptic  bevel  gear  is  the  invention  of  Pro- 
fessor MacCord. 

The  spherical  ellipse  cannot  be  drawn  by 


the  trammel  method  of  (138),  and  therefore 
the  method  of  spacing  of  (142),  as  well  as 


Elliptic  bevel  gears 

Fig.  141. 

the  gear  cutting  machine  of  (150),  does  not 
apply. 


84 


Elliptic    Calculations. 


153. — MATHEMATICAL  TREATMENT. 


If  the  major  semi-axis  is  a,  and  the  minor 
semi-  axis  is  b,  the  equation  of  the  curve  from 
the  origin  at  G  is 

a*  y*  -f&»z»=.  a8  b*, 
the  major  axis  being  the  axis  of  X. 

The  distance  GF  from  the  center  to  the 
focus  will  be 
c  = 


_&2  =  a  ^  i  _ 

in  which  TZ,  is  the  ratio  of  axes  =  —  . 

a 

The  radius  of  curvature  at  the  major  apex 

is  —  ,  and  that  at  the  minor  apex  is  —  . 
a  b 

There  is  no  practicable  formula  for  the  recti- 
fication of  the  curve,  as  the  length  is  express- 
ible only  by  a  series. 

The  special  spacing  method  of  (142)  is  true 
only  at  the  instant  of  passing  either  apex, 
for  the  tracing  point  describes  half  the  arc 
described  by  the  line  of  the  bar  on  the  index 
circle  only  when  the  bar  is  at  right  angles 
with  the  curve.  The  error  will  be  at  its 
maximum  when  the  bar  is  at  the  maximum 
angle  with  the  normal,  which  is  at  about  an 
angle  of  forty-five  degrees  with  the  major 
axis.  The  difference  between  an  ordinary 
tooth  space  at  the  major  apex,  and  that 
at  the  minor  apex,  is  very  minute.  A  very 
careful  calculation  of  the  length  of  the  chord 
of  a  gear  of  seventy-two  teeth,  and  eight  and 
ten  inch  axes,  gave  a  chord  of  .41433"  at  the 
major  apex,  a  chord  of  .41495"  at  45°  for  the 
maximum,  and  a  chord  of  .41441"  at  the 
minor  apex.  The  difference  between  the 
chords  at  the  apices  is  .00008",  but  as  the  cur- 


vature at  the  major  apex  is  greater  than  at  the 
minor  apex,  the  difference  between  the  arcs 
would  be  less,  perhaps  not  over  .00004". 
The  ratio  of  speeds  (148),  is 


/JL+jv^-^lY 
\  i  -  vi  —  K*  i 


The  ratio  of  quick  return  being  given  as 
qr,  the  value  of  n  is 

n  =  y  2  V  rf8  -f  d*  —  3d8, 
in  which  <Z  =  ton.  /-  18°   X° 


When  the  gears  are  in  a  train,  there  seems 
to  be  no  simple  method  for  computing  the 
ratio  of  axes  to  produce  a  given  quick 
return,  but,  when  the  ratio  is  given,  the 
quick  return  for  each  gear  can  be  computed 
best  by  trial  and  error  with  the  formula 
sin.  (M  —  N) 


sin.  M+sin.  N  " 
in  which  M  is  any  known  angle  b,  Fig.  136, 
and  -ZV  is  the  angle  b  for  the  next  following 
gear  in  the  train.  Thus,  assuming  n  =  .98, 
and  MI  =  90°,  we  find  N^  =  67°  28'.  Then 
putting  J/8  =  67°  28',  we  find  JT,  =  48°  5'. 
Knowing  the  angles,  we  compute  the 
quick  return  ratio  from 

' 


which,  for  n  =  .98  gives  qr  for  two  gears 
equal  to  1.66,  and  for  three  gears  equal  to 
2.74.  The  graphical  process  of  Fig.  136 
should  first  be  employed  to  fix  the  angles 
approximately. 


8.    THE    BEVEL    GEAR. 


154. — THE   BEVEL   GEAR. 


The  theory  of  the  bevel  gear  cannot  be 
properly  represented,  and  can  be  studied 
only  with  the  greatest  difficulty,  upon  a 
plane  surface.  It  is  essentially  spherical  in 
nature,  and  should  be  shown  upon  a  spheri- 
cal surface,  as  in  Figs.  143  and  144. 

This  is  best  done  upon  a  spherometer, 
which  is  simply  a  painted  sphere  fitted  in  a 
ring.  The  sphere  rests  upon  a  support,  so 
that  the  ring  coincides  with  a  great  circle 
upon  it,  and  the  ring  is  graduated  to  360°. 
A  very  roughly  made  wooden  sphere  and 
plain  ring  will  be  found  to  answer  the  gen- 


eral purpose  very  well,  and  should  be  pro- 
vided if  the  study  of  the  bevel  gear  is 
seriously  intended.  If  painted,  ink  marks 
can  be  scrubbed  off,  and  pencil  marks  re- 
moved with  a  rubber. 

The  mathematical  treatment  is  unapproach- 
able without  a  knowledge  of  the  common 
principles  of  spherical  trigonometry. 

A  wide,  interesting,  and  difficult  field  of 
study  is  offered,  but  space  will  permit  but  a 
brief  examination  of  the  more  prominent 
and  practical  points.  A  careful  examination 
would  require  ten  times  the  available  space. 


155. — THE  GENERAL   THEORY. 


When  thus  represented  upon  the  spherical 
surface,  the  theory  of  the  bevel  gear  is  so 
similar  to  that  of  the  spur  gear,  as  repre- 
sented upon  a  plane  surface,  that  any  de- 
tailed description  would  be  mostly  a  repeti- 
tion of  what  has  already  been  stated. 

All  straight  lines  of  the  spur  theory  are 
represented  by  great  circles,  the  crown  gear 
being  the  rack  among  bevel  gears,  and  all 
distances  are  measured  in  degrees. 

Irregular  pitch  lines  and  multilobes  are 
managed  substantially  as  for  spur  gearing. 
The  elliptic  bevel  gear  has  been  described  in 
connection  with  elliptic  spur  gears  (152). 

The  tooth  surfaces  of  the  bevel  gear  arc 
generally  formed  by  drawing  straight  lines 
from  the  spherical  outline  to  the  center  of  the 
sphere,  as  in  Figs.  143  and  144,  the  pitch  lines 
and  tooth  outlines  being  the  bases  of  cones 
with  a  common  apex. 

When  limited  in  width,  as  is  usually  the 
case,  it  is  by  a  sphere  concentric  with  the 
outside  sphere,  so  that  a  spherical  shell  is 
formed. 

These  concentric  spherical  shells  can  be 
moved  on  their  axes  to  form  twisted  and 
spiral  teeth,  Fig.  142,  precisely  as  described 
for  spur  gears  (99). 


The  molding  process  of  (27)  will  apply  per- 
fectly, but  it  has  but  one  practical  applica- 
tion. 


Fig.  142. 


Twisted  bevel  gear. 


The  planing  process  of  (28)  will  fail,  for 
practical  purposes,  except  for  one  particular 
form  of  tooth,  because  the  shape  of  the  cut- 
ting tool  cannot  in  the  general  case  be 


Involute  Bevel   Gears. 


changed  in  form  as  it  approaches  the  apex, 
and  therefore  the  tooth  will  not  be  conical. 

The  planing  process  of  (29)  will  apply  per- 
fectly, the  strokes  of  the  tool  being  radial, 
and  on  this  method  we  must  depend  for  the 
accurate  construction  of  all  forms  of  bevel 
gear  teeth  except  the  octoid  and  the  pin 
tooth. 

As  the  diameter  of  the  sphere  is  increased, 
the  radii  become  more  nearly  parallel,  until, 
when  the  diameter  is  infinite,  they  are  paral- 


lel. Therefore  the  spur  gear  is  a  particular 
case  of  the  bevel  gear,  and  all  formulae  and 
processes  that  apply  to  the  bevel  gear  will 
apply  to  the  spur  gear  if  the  diameter  of  the 
sphere  is  made  infinite.  The  most  scientific 
method  of  study  would  be  to  develop  the 
theory  of  the  bevel  gear,  and  from  that  pro- 
ceed to  that  of  the  spur  gear,  but  such  a 
method  would  be  difficult  to  clearly  carry 
out,  and  is  best  abandoned  for  the  more  con- 
fined process  here  adopted. 


156. — PARTICULAR  FORMS  OF  BEVEL  TEETH. 


As  in  the  case  of  spur  gearing,  there  can 
be  an  infinite  number  of  tooth  curves  for 
bevel  gearing  (31),  each  form  having  its  own 
line  of  action,  but  as  there  are  only  four 
forms  that  are  available  for  practical  use  by 
means  of  simple  processes  of  construction, 
our  attention  will  be  confined  to  them. 

These  four  particular  forms  are,  first,  the 


involute  tooth,  having  a  great  circle  line  of 
action;  second,  the  cycloidal  tooth,  having  a 
circular  line  of  action;  third,  the  octoid 
tooth,  having  a  plane  crown  tooth,  and  a 
"figure  eight"  line  of  action;  and,  fourth, 
the  pin  tooth,  for  which  one  gear  of  a  pair 
has  teeth  in  the  form  of  round  pins. 


157. — THE  INVOLUTE  BEVEL  TOOTH. 


The  spherical  involute  must  be  studied  as 
a  whole  if  its  form  is  to  be  clearly  seen. 

Its  definition  is  that  it  is  the  tooth  curve 
having  a  great  circle  for  a  line  of  action.    In 
Fig.  143  the  great  circle  line  of  action  la  ex- 
tends around  the  sphere  at  an  angle  with  the 
crown  pitch  line  pi,  and  it  is  tangent  to 
two  base  lines  bl  and  bl' ' ,  that  are  paral- 
lel with  the  crown  line. 

The  most  convenient  method  of  draw- 
ing the  tooth  curve  is  by  rolling  the  line 
of  action  on  the  base  line,  while  a  point 
in  it  describes  the  curve  on  the  surface 
of  the  sphere.  The  equivalent  graphical 
process  is  to  step  along  the  base  line 
and  any  two  tangent  great  circles,  from 
any  point  on  the  curve  to  any  desired 
point. 

It  will  take  the  form  shown  by  the 
dotted  lines ;  rising  at  right  angles  to  the 
base  line,  it  curves  until  the  crown 
line  is  reached,  there  reversing  its  curva- 
ture and  bending  the  other  way  until  it 
meets  the  other  base  line.  At  the  base 
line  it  has  a  cusp,  and  rises  from  it  to 
repeat  the  same  course  indefinitely. 


Fig.  143  shows  a  crown  gear  or  rack.  The 
pitch  line  is  the  great  circle  pi.  The  line 
of  centers  cOCis  a  great  circle  at  right  angles 
with  the  crown  line  pi.  The  line  of  action 
is  the  great  circle  la  set  at  a  given  angle  of 
obliquity  with  the  crown  line.  The  base 

Fit/.  143. 

The  involtite  Tooth 


Bilgram  Bevel   Gears. 


87 


circles  are  the  small  circles  bl  and  bl'.  The 
spherical  involutes  have  the  same  property  of 
adjustability  as  have  the  spur  involutes,  the 


motion  being  confined  to  the  sphere,  and  there- 
fore the  gears  are  adjustable  as  to  their  shaft 
angle,  the  apex  remaining  common  to  both. 


158. — THE  CYCLOIDAL  BEVEL  TOOTH. 

The  definition  of  the  cycloidal  tooth  is  I  flank  formed  by  a  roller  of  half  the  angular 
that  it  is  that  form  which  has  a  circular  line   diameter  of  the  gear  being  nearly  but  not 


of  action. 

The  rolled  curve  method  of  treatment  (32) 
applies,  and  is  the  best  means  of  studying 
the  curve. 

There  is  no  gear  with  radial  flanks,  the 


exactly  a  plane. 

The  theory  differs  so  little  from  that  of 
the  spur  gear,  that  but  little  of  interest  can 
be  found,  and  the  curve  will  not  be  consid- 
ered further. 


159. — THE  OCTOID  BEVEL  TOOTH. 


The  definition  of  this  tooth  system  is  that 
it  is  the  conjugate  system  derived  from  the 
crown  gear  having  great  circle  odontoids. 

In  Fig.  144  the  crown  gear  has  plane 
teeth  cutting  the  sphere  in  great  circles, 
mOn,  while  a  pinion  would  have  convex 
tooth  curves  conjugate  to  the  great  cir- 
cles of  the  crown  tooth. 

The  line  of  action,  from  which  the 
tooth  derives  its  name,  is  the  peculiar 
' '  figure  eight "  curve  la,  which  is  at  right 
angles  to  the  tooth  curve  at  the  crown 
line  pi,  and  tangent  to  the  polar  circles 
JS  and  &,  to  which  the  great  circle  crown 
odontoids  are  also  tangent. 

This  tooth  owes  its  existence  to  the 
fact  that  it  is  the  only  known  tooth,  and 
probably  the  only  possible  tooth,  that 
can  be  practically  formed  by  the  mold- 
ing planing  process  of  (28).*  The  cutting 
edge  of  the  tool  being  straight,  no 
change  is  required  while  it  is  in  motion, 
except  in  its  position,  and  that  is  accom- 
plished by  giving  it  a  motion  in  such 
a  direction  that  its  corner  moves  in  the  radial 
line  of  the  corner  of  the  bottom  of  the  tooth 
space. 

The  octoid  tooth,  together  with  an  ingeni- 

*  Since  this  statement  was  made,  another  bevel 
practically  constructed  by  the  process  of  (28). 


ous  machine  for  planing  it,  was  invented  by 
Hugo  Bilgram,  but  it  has  always  been  mis- 
taken for  the  very  similar  true  involute  tooth. 


The  Octoid  Tooth 

Fig.  144. 

Bilgram's  machine  is  described  in  the 
AMERICAN  MACHINIST  for  May  9th,  1885, 
and  in  the  Journal  of  the  Franklin  Institute 
for  August,  1886. 

tooth,   the   "  planoid  "    tooth,   has  been  invented  and 


160. — THE   PIN   BEVEL   TOOTH. 


If  the  tooth  of  one  gear  of  a  pair  is  a  coni- 
cal pin,  Fig.  145,  with  apex  at  the  center  of 
the  sphere,  that  of  the  other  will  be  conju- 


gate to  it,  and  the  combination  deserves 
notice  because  it  is  one  of  the  few  forms  that 
are  easily  constructed.  It  may  be  said  that 


88 


TredgolcTs  Method. 


Tig.  145. 


its  practical  construction  is  simpler  and  easier 
than  that  of  any  other  form  of  bevel  gear 
tooth  except  the  skew  pin  tooth  of  (180). 

The  tooth  is 
preferably,  but 
not  necessarily, 
of  the  conical 
form,  for  other 
forms  of  circu- 
lar pins  would 
serve  the  theo- 
retical p  u  r  - 
pose. 
Its  theory  is, 


Pin  bevel  gears. 


in  the  main,  the  same  as  that  of  the  spur  pin 


tooth.  It  has  the  same  troublesome  cusp, 
which  can  be  avoided  in  the  same  way,  b^r 
setting  the  center  of  ihe  pin  back  from  the 
pitch  line. 

It  is  the  only  known  form  of  tooth  that 
can  be  formed  in  a  practical  manner  by  the 
molding  process  of  (27).  If  the  cutting  tool 
is  a  conical  mill,  it  will  form  the  conjugate 
tooth  while  the  two  pitch  wheels  are  rolled 
together. 

The  pins  may  be  mounted  on  bearings  at 
their  ends, '  forming  roller  teeth.  They 
would  be  weak,  but  would  run  with  the 
least  possible  friction,  all  the  rubbing  friction, 
being  confined  to  the  bearings. 


161. — TREDQOLD'S  APPROXIMATION. 


The  construction  of  the  true  bevel  gear 
tooth  curve  upon  the  true  spherical  surface  is 
impracticable  with  the-means  in  ordinary  use, 
and  the  true  method  of  computation  by  means 
of  spherical  trigonometry  is  equally  unfitted 
for  common  use.  But,  by  adopting  Tred- 
gold's  approximate  method  the  difficulties  can 
be  overcome. 

By  this  method  the  tooth  curves  are  drawn, 
not  on  the  true  spherical  surface,  but,  as  in 
Fig.  146,  on  cones  A  and  B  drawn  tangent 
to  the  sphere  at  the  pitch  lines  of  the  gears. 
The  cones  are  then  rolled  out  on  a  plane  sur- 
face, and  the  gear  teeth  drawn  upon  them 
precisely  as  for  spur  gears  of  the  same  pitch 
diameter. 

Practically  correct  tooth  curves  could  thus 
be  drawn  on  the  spherical  surface  by  cutting 
the  teeth  to  shape,  and  bending  them  down 
to  scribe  around  them,  but  in  practice  the 
back  rims  of  the  gears  are  shaped  to  the  tan- 
gent cones  so  that  the  teeth  lie  directly  upon 
the  conical  surface. 

This  method  is  called  approximate,  but  its 
real  error  would  be  difficult  to  determine, 
and  is  certainly  not  as  great  as  the  inevitable 
errors  of  workmanship  of  any  graphical  pro- 
cess. The  tooth  outline  drawn  by  it  upon 
the  spherical  surface  may  be  considerably 
different  from  that  which  would  be  drawn 
directly  upon  it,  but  it  does  not  follow  that 
it  is  therefore  incorrect.  The  only  require- 
ment is  that  the  engaging  curves  shall  be 


Fig 


Tredffold's  method. 

conjugate  odontoids,  and  it  is  a  matter  of 
very  small  consequence  whether  or  not  the 
curve  on  the  sphere  is  the  same  kind  of  curve 
as  that  upon  the  cone.  If  the  true  plane  in- 
volute curve  is  drawn  upon  the  developed 
cone,  the  corresponding  curve  on  the  sphere 
will  not  be  an  exact  spherical  involute,  but  its 
divergence  from  some  true  odontoidal  shape 
must  be  minute,  even  when  the  teeth  are  very 
large  indeed.  In  ordinary  cases  it  cannot  be 
sufficient  to  affect  materially  the  constancy 
of  the  velocity  ratio.  What  is  sometimes 
given  as  its  error  is  mostly  the  "difference  in 
shape"  between  the  plane  and  the  spherical 
teeth. 


Drafting  Bevel   Gears. 


89 


162. — DRAFTING  THE  BEVEL  GEAR. 


The  practical  application  of  Tredgold's 
method  is  illustrated  by  Fig.  147. 

Draw  the  axes  CA  and  CB  at  the  given 
shaft  angle  ACE.  Lay  off  the  given  pitch 
radii  a  and  b,  and  draw  the  lines  c  and  d  in- 
tersecting at  the  pitch  point  0.  Dra  the 
center  line  OC,  and  lay  off  the  face  Of. 

The  pitch  diameters  are  ON  and  OM, 
and  NCO  and  MCO  are  the  pitch  cones. 

Draw  the  back  rim  line  02)  at  right  an- 
gles with  the  center  line,  lay  off  the  addenda 
Oe  and  Og,  and  the  clearance  gh.  Draw  the 
front  rim  line  parallel  to  the  back  rim  line. 


The  center  angle  is  X,  the  face  increment 
I  is  F,  and  W  is  the  face  angle.  The  cutting 
I  decrement  is  J,  and  T  is  the  cutting  angle. 
!  Twice  the  distance  mn  is  the  diameter  incre- 
!  ment,  and  em  is  the  outside  diameter. 

The  pitch  radius  of  the  Tredgold  back 
cone  is  OD,  and  the  figure  shows  the  con- 
struction of  the  gear  teeth  on  this  cone 
developed.  The  teeth  are  represented  as 
drawn  upon  the  figure,  but  it  is  better  to  use 
a  separate  sheet.  The  odontograph  should 
be  used,  calculating  the  number  of  teeth  in 
the  full  circle  of  the  developed  cone. 


Drafting  the  bevel  gear* 


163. — THE  BEVEL  GEAR  CHART. 


The  drafting  of  the  bevel  gear  blanks  by 
means  of  the  method  of  (162)  is  simple,  but 
the  method  requires  drafting  instruments, 
not  always  at  hand,  as  well  as  the  ability  to 
use  them  accurately.  The  drawing  must  be 
carefully  made,  to  give  correct  results,  par- 
ticularly when  the  gears  are  small.  After 
the  drawing  is  made  the  various  angles  and 


diameters  must  be  taken  off  for  use  at  the 
lathe,  and  that  is  by  no  means  a  simple  mat- 
ter. 

So  great  are  the  practical  difficulties  that 
any  one  who  has  a  knowledge  of  simple  arith- 
metic will  find  it  not  only  easier,  but  more 
accurate  to  use  the  chart  and  method  by 
means  of  the  following  rules. 


90 

THE    BEVEL    GEAR    CHART. 


Shafts  at  90° 
I'roportion. 

Center 
Angle. 

Angle  Incr. 
Div.  by  Teeth 

p  S 

Shafts  at  00° 
Proportion. 

Center 
Angle. 

Angle  Incr. 
Div.  by  Teeth 

II 

.10      1—10 

5.72 

11 

2.00 

10.00 

10—1 

84.28 

114 

.20 

.11      1—9 

6.33 

13 

2.00 

9.00 

9—1 

83.67 

114 

.22 

.13      1—  8  |       7.12 

14 

1.99 

8.00 

8—1 

82.88 

113 

.25 

.14 
.17 

1—  7 

8.13 

16 

1.98 

7.00 

7—1 

81.87 

113 

.28 

1—  6 

9.47 

19 

1.97 

6.00 

6—1 

80.53 

113 

.33 

.20      1—  5  |     11.32 

23 

1.96 

5.00 

5—1 

78.68 

112 

.89 

.22      2—  9 

12.53 

25 

1.95 

4.50 

9—2 

77.47 

111 

.43 

.25      1—  4 

14.03 

28 

1.94 

4.00 

4—1 

75  97 

111 

.29      2—  7 

15.95          32 

1  92 

3.50 

7—2 

74.05 

110 

.55 

.30      3—10 

16.70 

33 

1.92 

3.33 

10—3 

73.30 

109 

.58 

.33      1—  3 

18.44 

36 

1.90 

3.00 

3—1 

71.57 

109 

.63 

.38      3—  8 

20.55 

40       1.87 

2  67 

8—3 

69.45 

107 

.70 

.40      2—5 

21.80 

43       1.86 

2.50 
2.33 

5—2 

68.20 

106 

.74 

.43 

3—  7 

23.20 

45       1.84 

7—3 

66.80 

105 

.79 

.44 

4—  9 

23.97 

46       1.83 

2.25 

9—4 

66.03 

104 

.81 

.50 

1—  2 

26.57 

51       1.79 

2.00        2—1 

63.43 

103 

.89 

.56 

5—  9 

29.05 

56       1.74 

1.80 

9—5 

60.95 

101 

.97 

.57 

4—  7       29.75 

57       1.74 

1.75 

7—4 

60.25 

99 

.99 

.60 

3_  5       30.97 

59       1  72 

1.67 

5—3 

59.03 

98 

1.03 

.63 

5—  8       32.00 

61       1.69 

1.60 

8—5 

58.00 

97 

1.06 

.67 

2—  3       33.68 

64       1.66 

1.50 

3—2 

56.32 

95  |   1.11 

.70 

7—10       34.99 

66       1.64 

1.43 

10—7 

55.00 

94 

1.15 

.71 

5—  7       35.53 

67       1.63 

1.40 

7—5 

54.47 

93 

1.16 

.75 

3_  4       36.87 

69 

1.60 

1.33 

4—3 

53.13 

92 

1.20 

.78 

7—  9       37.87 

70 

1.58 

1.29 

9—7 

52.13 

91 

1  22 

.80 

4—  5       38.67 

72 

1.56 

1.25 

5—4 

51.33 

90 

1.25 

.83 

5—  6       39.80 

73 

1.54 

1.20 

6—5 

50.20 

88 

1.28 

.86 

6—  7       40.60 

75 

1.52 

1.17 

7—6 

49.40 

87 

.88 

7—  8       41.18 

76 

1.50 

1.14 

8—7 

48.82 

86 

1.32 

.89 

8—  9       41.63 

76 

1.49 

1.13 

9—8 

48.37 

86 

1.33 

.90 

9—10 

41.98 

77 

1.49 

1.11 

10—9 

48.02 

85 

1.34 

1.00 

1—  1 

45.00 

81 

1.41 

1.00 

1—1 

45.00 

81 

1.41 

Specimen    Chart    Calculations. 


91 


FIG.  148. 
SAMPLE  COMPUTATION. 


SHAFTS  AT 
A   RIGHT  ANGLE. 


Pitch  =  3      Prop.  =  7  —  5 

Shaft  ang.  90j 

Teeth  =  42)  93    (2.22  = 
84       .37  + 

face  incr.          | 
* 

90   2.59  = 

84 

60 

cut  deer. 

Center  angles  =  54.47 
_j_  incr                    2  22 

35.53 
2  22 

Face  angles  56.69 

37.75 

Center  angles  =  54.47 
—  deer  2.59 

35.53 
2.59 

Cut  angles  51.88 

32.1)4 

Pitch  =3)1.16 

3)  1.63 

Diam.  incr.  =        .39 
-\-  p.  diams.  =    14. 

.54 
10. 

o.   diams.  =    14.39 

10.54 

FIG.  149. 
SAMPLE  COMPUTATION. 


SHAFTS  AT 
ANY    ANGLE. 


Pitch  =  5       Prc 

>p.  =  X 

Shaft  ang.  52.8 

Teeth  =  20) 

66    (3.30  = 

.55  + 

3.85  = 

face  incr. 
i 

cut  deer. 

Center  angles 

' 

=  35.80 
3.30 

17.00 
3.30 

Face  angles 

=   39.10 

20.30 

Center  angles 
••  deer     .... 

=  35.80 
...    3.85 

17.00 
3.85 

Cut    angles 

=   31.95 

13.15 

Pitch  =  5) 

1.66 

5)  1.91 

Diam.  incr  
-j-  p.  diams  

.33 

4. 

.38 

2. 

o.  diams.  .  .  . 

4.33 

2.38 

92 


Chords  of  Angles. 


164. — SHAFTS  AT  BIGHT  ANGLES. 


1st. — Divide  the  pitch  diameter  by  that  of 
the  other  gear  of  the  pair,  or  else  the  number 
of  teeth  by  that  of  the  other  gear,  to  get  the 
proportion.  Enter  the  table  by  means  of  the 
proportion.  All  numbers  for  that  pair  will 
be  found  on  the  same  horizontal  line  in  the 
two  columns. 

2d.— The  center  angles  are  given  directly 
by  the  table  at  the  proper  proportion. 

3d. — Divide  the  tabular  angle  increment 
by  the  number  of  teeth  in  the  gear,  to  get 
the  angle  increment.  This  need  be  done  for 
but  one  gear  of  a  pair,  as  the  increment 
is  the  same  for  both. 


4th. — Add  the  angle  increment  to  the  cen- 
ter angle,  to  get  the  face  angle. 

5th.  —Increase  the  angle  increment  by  one- 
sixth  of  itself,  to  get  the  cutting  decrement, 
and  subtract  this  decrement  from  the  center 
angle,  to  get  the  cutting  angle. 

6th. — Divide  the  tabular  diameter  incre- 
ment by  the  diametral  pitch,  to  get  the 
diameter  increment,  and  add  that  to  the  pitch 
diameter,  to  get  the  outside  diameter. 

Fig.  148  is  a  sample  computation  for  shafts 
at  right  angles. 


165.— SHAFTS  NOT 

The  table  cannot  be  entered  by  means  of 
the  proportion,  and  the  numbers  for  the  two 
gears  of  the  pair  will  not  be  found  on  the  same 
horizontal  line,  and  it  will  be  necessary  to 
determine  the  center  angles. 

As  in  Fig.  147,  draw  the  axes,  at  the  given 
shaft  angle,  and  find  the  center  angles,  by  the 
method  described  in  (162). 

Then  enter  the  table,  for  each  gear  by 
itself,  by  means  of  the  center  angles,  and 
proceed  as  for  shafts  at  right  angles.  The 
angle  increment  and  decrement  is  the  same 
for  both  gears  of  a  pair. 

Fig.  149  is  a  sample  computation  applied 


AT  BIGHT  ANGLES. 

to  the  case  of  Fig.  147,  the  center  angles  be- 
ing found  by  means  of  the  table  of  chords. 
If  preferred,  the  center  angles  can  be  found 
by  means  of  the  formula, 

sin.  8 


tan.   C  = 


in  which  C  is  the  center  angle  of  the  gear,  P 
is  the  proportion  found  by  dividing  the  num- 
ber of  the  teeth  in  the  gear  by  the  number  in 
the  other  gear,  and  8  is  the  shaft  angle. 
Having  found  one  center  angle,  subtract  it 
from  the  shaft  angle  to  get  the  other  center 
angle. 


166. — THE  TABLE  OF  CHOBD8  AT  SIX  INCHES. 


When  the  lathesman  is  provided  with  a 
graduated  compound  rest  which  feeds  the 
tool  at  any  angle,  nothing  but  the  computa- 
tion is  required;  but  when  there  is  nothing 
but  the  common  square  feed,  the  faces  must 
be  scraped  with  a  broad  tool.  A  templet  for 
guiding  the  work  can  easily  be  made  by 
means  of  the  table  of  chords  at  six  inches. 

To  lay  out  a  given  angle,  draw  an  arc 
with  a  radius  of  six  inches,  draw  a  chord  of 
the  length  given  by  the  table  for  the  angle, 


and  then  draw  the  sides  oc  and  ob  of  the 
angle  boc,  Fig.  150. 

For  tenths  of  a  degree  use  the  small  tables. 
The  chord  of  37.5°  is  3.81  +  .05  =  3.86 
inches. 

Fig.  151  shows  the  manner  of  using  the 
angle  templet  at  the  lathe. 

This  table  of  chords  is  very  convenient  for 
many  purposes  not  connected  with  gearing, 
and  it  is  more  accurate  than  the  common 
horn  or  paper  protractor. 


167.— BILGBAM'S  CHABT. 


A  graphical  method  for  determining  the 
angle  and  diameter  increments,  the  invention 
of  Hugo  Bilgram,  is  described  in  the  AMEBI- 
CAN  MACHINIST  for  November  10,  1883.  It 


determines  the  required  values  by  the  inter- 
sections of  lines  and  circles,  and  requires  no 
computation. 


Chords  of  Angles. 


Chord  of  «»*  angle 

Fig.  150. 


Using  the  templet 


TABLE    OF    CHORDS    OF    ANGLES, 

AT  RADIUS   OF    SIX  INCHES. 


Degrees. 

Chord. 

Tenths. 

Degrees. 

Chord. 

Tenths. 

Degrees. 

Chord. 

Tenths. 

1 
2 
3 
4 
5 

.10 
.20 
.31 
.42 
.52 

31 
32 
33 
34 
35 

3.20 
3.31 
3.41 
3.51 
3.61 

61 
62 
63 
64 
65 

6.10 
6.19 
6.28 
6.36 
6.45 

6 

7 
8 
9 
10 

.62 
.73 
.84 
.94 
1.04 

36 
37 
38 
39 
40 

3.71 
3.81 
3.91 
4.01 
4.10 

66 
67 
68 
69 
70 

6.54 
6.62 
6.71 
6.80 
6.89 

11 
12 
13 
14 
15 

.15 
.26 
.36 
.46 
.57 

.1—  .01 
.2  —  .02 
.3—  .03 

41 
42 
43 
44 
45 

4  20 
4.30 
4.40 
4.50 
4.60 

.1—  .01 
.2—  .02 
.3—  .03 

71 

72 
73 
74 
75 

6.97 
7.06 
7.14 
7.22 
7.31 

.1—  .01 
.2—  .02 
3      02 

16 
17 

18 
19 
20 

.67' 

.77 
.87 
98 
2.08 

.4—  .04 
.5—  .05 
.6—  .06 
.7—  .07 
.8—  .08 

46 
47 

48 
49 
50 

4.69 
4.79 
4.88 
4.98 
5.08 

.4—  .04 
.5—  .05 
.6—  .05 
.7—  .06 

.8—  .07 

76 
77 
78 
79 
80 

7.39 
7.47 
7.55 
7.63 
7.71 

.4—  .03 
.5—  .04 
.6—  .05 
.7—  .06 
.8—  .06 

21 
22 
23 
24 
25 

2.18 
2.29 
2.39 
2.49 
2.59 

.9  —  .09 

51 
52 
53 
54 
55 

5.17 
5.26 
5.35 
5.45 
5  54 

.9  —  .08 

81 

82 
83 
84 
85 

7.79 
7.87 
7.95 
8.03 
8.11 

.9—  .07 

26 
27 

28 
29 
30 

2.70 
2.80 
2.90 
3.00 
3.10 

56 

57 
58 
59 
60 

5.63 
5.72 

5.82 
5.91 
600 

86 

87 
88 
89 
90 

8.18 
8.26 
8.34 
8.41 
8.48 

Templet  Planer. 


168. — ROTARY  CUT  BEVEL  TEETH. 

The  most  common  method  of  forming  the  !  ehape  of  the  tooth  changes,  while  that  of  the 
teeth  of  the  bevel  gear  is  by  cutting  them   cutter  is  invariable.     Therefore    the    result 


from  the  solid  blank  by  the  use  of  the  com- 
mon rotary  cutter. 

The  cutter  should  be  shaped  to  cut  the 
tooth  of  the  correct  shape  at  the  large  end,  and 
the  small  end  must  be  shaped  either  by  an- 
other cut  with  a  different  cutter,  or  with  a  file. 

It  is  impossible  to  cut  the  tooth  correctly 
at  both  ends,  for  the  simple  reason  that  the 


must  always  be  an  approximation  depending 
upon  the  personal  skill  and  experience  of  the 
workman.  It  is  a  too  common  practice  to 
make  the  teeth  fit  at  the  large  ends,  and  to 
increase  the  depth  of  the  tooth  toward  the 
point,  so  that  the  teeth  will  pass  without 
filing,  but  such  teeth  can  be  in  working  con- 
tact only  at  the  large  ends. 


). — THE  TEMPLET  GEAR  PLANER. 


The  most  common  method  of  planing  the 
teeth  of  bevel  gears  is  by  means  of  devices 
adapted  to  guide  the  tool  by  a  templet  that 
has  previously  been  shaped,  as  nearly  as 
may  be,  to  the  true  curve.  The  arm  that 
carries  the  tool  is  hung  by  a  universal  joint 
at  the  apex  of  the  gear,  so  that  all  of  its 
strokes  are  radial,  and  a  finger  placed  in  the 
line  of  the  stroke  of  the  cutting  point  of  the 
tool  is  held  against  the  templet.  There  are 
many  different  arrangements  for  the  purpose, 
but  they  are  all  founded  on  the  same  princi- 
ples, and  differ  only  as  to  details. 


The  invention  of  the  templet  gear  planer  is 
commonly  credited  to  George  H.  Corliss, 
who  patented  it  in  1849,  and  was  the  first  to 
use  it  in  this  country.  But  it  was  patented 
in  France,  by  Glavet,  in  1829,  and  may  be 
even  older. 

It  is  largely  used  for  planing  the  teeth  of 
heavy  mill  gearing,  but  has  not  been,  and 
cannot  be,  profitably  applied  to  common 
small  gear  work.  Its  product  is,  in  any  case, 
superior  to  the  rough  cast  tooth,  but  its  accu- 
racy is  dependent  on  that  of  the  templet,  and 
is  therefore  dependent  on  personal  skill. 


9.    THE    SKEW     BEVEL    OEAR. 


170. — THE   SKEW  BEVEL   GEAR. 


When  a  pair  of  shafts  are  not  parallel,  and 
do  not  intersect,  they  are  said  to  be  askew 
with  each  other,  and  they  may  be  connected 
by  a  pair  of  skew  bevel  gears,  having 
straight  teeth,  which  bear  on  each  other 
along  a  straight  line.  Such  gears  are  to  be 
carefully  distinguished  from  spiral  gears, 
used  for  the  same  purpose  but  having  spiral 
teeth  bearing  on  each  other  at  a  single  point 
only. 

We  will  endeavor  to  describe  the  skew 
bevel  gear  so  that  its  general  nature  can  be 
understood,  but  it  is  impossible  to  do  so  in 
simple  language.  It  is  the  most  difficult  ob- 
ject connected  with  the  subject.  The  theory 
cannot  even  be  considered  as  yet  settled,  for 
writers  upon  theoretical  mechanism  do  not 
agree  upon  it,  and  there  are  points  yet  in 
controversy. 

In  the  theory  of  the  bevel  gear  the  surface 
of  reference  is  the  spherical  surface  upon 
which  the  tooth  outlines  are  drawn,  and  upon 


which  the  laws  of  their  action  may  be 
studied,  for  spheres  of  reference  of  two  sep- 
arate gears  may  be  made  to  coincide  so  that 
the  lines  upon  one  will  come  in  contact  with 
those  upon  the  other.  For  the  spur  gear,  the 
spheres  become  planes  and  the  process  is  the 
same.  But  for  the  skew  bevel  gear  there  is 
no  analogous  process,  for  it  is  impossible  to 
imagine  a  surface  of  such  a  nature  that  it  can 
be  made  to  coincide  with  a  similar  surface 
when  both  are  attached  to  revolving  askew 
shafts.  There  are  spiral  surfaces  which  will 
approximately  coincide,  and  are  analogous  to 
the  Tredgold  tangent  cones  of  bevel  gears 
(161),  but  any  tooth  action  developed  upon 
such  approximate  surf  aces  must,  of  necessity, 
be  not  only  approximate,  but  also  very  diffi- 
cult to  define  and  formulate. 

Of  all  the  skew  tooth  surfaces  that  have 
been  proposed,  there  is  but  one,  the  Olivier 
involute  spiraloid,  that  can  be  proved  to  be 
theoretically  correct. 


171. — THE  HYPOID. 


The  pitch  surface  of  the  ?kew  bevel  gear 
is  the  surface  known  as  the  "  hyperboloid  of 
revolution,"  and  it  is  so  intimately  connected 
with  the  subject  that  it  must  be  thor- 
oughly understood  before  going  further. 
The  clumsy  name  may  be  abbreviated  to 
"hypoid." 

If  a  line  D,  Figs.  152  and  153,  called 
a  generatrix,  is  attached  to  a  revolving 
shaft  A,  so  that  it  revolves  with  it,  it 
will  develop  or  ' '  sweep  up  "  the  hypoid 
H  in  the  space  surrounding  the  shaft. 
A  section  of  the  surface  by  any  plane 
normal  to  the  axis  is  a  circle.  The  com- 
mon normal  to  the  generatrix  and  the 
axis  is  the  gorge  radius  G,  and  circular 
section  through  that  line  is  the  gorge 
circle.  A  section  by  a  plane  D,  Fig.  152, 
parallel  to  the  axis,  at  the  gorge  distance 
from  the  axis,  will  be  the  pair  of  straight 


lines  d  and  d ',  Fig.  153,  either  one  of  which 
is  an  element  of  the  surface,  and  will  form  it 
if  used  as  a  generatrix.  A  section  by  any 


sections. 

Fig.  152. 


Hyperbolic  sections. 

Fig.  153. 


other  plane  parallel  to  the  axis  will  be  a 
hyperbola,  to  which  the  elements  d  and  d' 


Rolling  Hypoids. 


are  assymptotes,  or  lines  which  the  curves 
continually  approach,  but  reach  only  at  in- 
finity. Fig.  153  shows  at  Q  the  hyperbolas 
cut  by  the  plane  Q  of  Fig.  152,  and  at  R 
those  cut  by  the  plane  R.  The  principal 
hyperbola  H  is  the  only  one  with  which 
we  are  concerned . 

The  hypoid  is  best  studied  as  projected 
upon  a  plane  parallel  to  the  axis,  as  in  Fig. 
154,  in  which  A  is  the  projection  of  the  axis, 
d  is  that  of  the  generatrix,  dGA  is  the  skew 
angle,  and  //is  the  principal  hyperbola. 

When  the  skew  angle  and  the  gorge  radius 
are  given,  the  hyperbola  is  easily  constructed 
by  points.  Any  line  ab  is  drawn  normal  to 
the  axis  and  the  gorge  distance  be=  Gg  is  laid 
off  from  b,  the  distance  ab  is  made  equal  to 
ec,  and  a  is  then  a  point  on  the  curve.  The 
curve  is  to  be  drawn  through  several  points 
thus  constructed. 


The  hyperbola. 

Fig.  154. 

To  draw  a  tangent  to  the  curve  at  any 
point  a,  draw  a  line  am  parallel  to  the 
assymptote  d,  lay  off  mn  equal  to  Gm,  and 
draw  the  tangent  an. 


172.  — THE   PITCH   HYPOIDS. 


The  utility  of  the  hypoid  as  the  pitch  sur- 
face of  the  skew  gear  depends  upon  the  pe- 
culiar property  that  any  number  of  such 
surfaces  will  roll  together,  and  drive  each 
other  by  frictional  contact  with  velocity  ratios 
in  the  proportions  of  the  sines  of  their  skew 
angles,  if  their  gorge  radii  are  in  the  propor- 
tions of  the  tangents  of  their  skew  angles. 

It  is  required  to  construct  a  pair  of  rolling 
hypoids  that  will  transmit  a  given  velocity 
ratio  between  two  shafts  that  are  set  at  a 
given  angle  with  each  other.  In  Fig.  155, 
A  and  .Bare  the  given  axes,  and  AGB  the 
given  shaft  angle.  The  directrix  D  is  to  be 
so  drawn  that  the  sines  of  the  skew  angles 
AGD  and  BGD  are  in  the  proportion  of  the 
given  velocity  ratio,  and  this  is  best  done  by 
drawing  lines  parallel  to  the  axes,  at  distances 
from  G  that  are  in  the  given  ratio,  and  drawing 
the  directrix  through  their  intersection  D. 

In  the  figure  the  axes  are  situated  one  over 
the  other  at  a  distance  GH  called  the  gorge 
distance,  and  the  directrix  D  is  situated  be- 
tween them  so  as  to  pass  through  the  gorge 
line  and  divide  the  gorge  distance  into  gorge 
radii,  #TFand  HW,  which  are  in  proportion 
to  the  tangents  of  the  skew  angles.  This  is 


Pitch  hypoids. 

Fig.  155. 


best  done  by  drawing  cd  normal  to  GD  in 
any  convenient  position,  laying  off  the  gorge 
distance  ce  at  any  convenient  angle  with  cd, 
and  drawing  de  and  gf  parallel  to  it;  cf  will 
be  the  goige  radius  GW  for  the  axis  GA, 


Rolling   Hypoids. 


97 


and  fe  will  be  the  gorge  radius  HW  for  the 
axis  OS. 

Then,  if  the  directrix,  thus  situated,  is  at- 
tached first  to  one  shaft  and  then  to  the 
oiher,  and  used  as  a  generatrix,  it  will  sweep 
up  a  pair  of  pitch  hypoids  that  will  be  in 
contact  at  the  directrix,  and  which  will  roll 
on  each  other. 

They  will  not  only  roll  on  each  other  in 
contact  at  the  directrix,  but  they  will  also 
have  a  sliding  motion  on  each  other  along 
that  line,  the  two  motions  combining  to  form 
a  resulting  motion  that  must  be  seen  to  be 
understood.  It  is  this  sliding  motion  that 
makes  all  the  difficulty  in  the  construction  of 
the  teeth,  for  they  must  be  so  constructed 
as  to  allow  it.  It  is  also  the  cause  of  the 
great  inefficiency  of  such  teeth  in  action,  for 
any  possible  form  must  have  a  lateral  sliding 
motion,  with  the  consequent  friction  and 
destruction. 

If  we  draw  two  diameters  mn  and  m'n' 
through  the  same  point  C  on  the  directrix, 
they  will  be  the  diameters  of  circles  that  will 
touch  each  other  while  revolving,  and  may 


be  called  pitch  circles.  If  they  are  thin,  and 
provided  with  teeth  in  the  given  velocity 
ratio,  they  will  drive  each  other  with  a  con- 
tact that  is  approximately  correct,  and  if 
there  are  several  pairs  of  such  thin  gears  set 
so  far  apart  that  they  do  not  interfere  with 
each  other,  they  will  serve  light  practical 
purposes  fairly  well. 

If  a  face  distance  CE  is  laid  off  on  the 
directrix  and  another  pair  of  pitch  circles 
constructed,  the  frustra  of  the  hypoids  in- 
cluded between  the  circles  may  be  called 
pitch  frustra,  and  they  will  roll  together  in, 
contact  at  the  directrix. 

It  is  to  be  noticed  that  the  pitch  diameters 
thus  determined  are  not,  as  in  spur  and  bevel 
gearing,  in  the  inverse  proportion  of  the 
velocity  ratio  of  the  axes,  and  therefore  if 
one  diameter  of  a  pair  of  skew  gears  to  have 
a  given  velocity  ratio  is  given,  the  other  must 
be  constructed.  When  the  skew  angles  are 
equal,  the  pitch  diameters  are  equal,  but 
otherwise  the  proportion  cannot  be  expressed 
in  simple  terms,  and  must  be  determined  by 
making  the  drawing. 


173. — THE  LOCUS  OF  AXES. 


The  rolling  hypoids  may  be  examined 
from  another  and  most  interesting  point  of 
view.  In  Fig.  156  the  gorge  line  G  is  nor- 
mal, and  the  directrix  D  is  parallel  to  the 
plane  of  the  figure.  The  plane  P  is  normal 
to  the  directrix,  and  below  is  a  front  view  of 
it.  On  the  plane  P  draw  any  straight  line  L 
through  the  directrix.  From  any  two  points 
a  and  b  on  this  line  draw  lines  A  and  B 
normal  to  the  gorge  line  G,  and  they  will 
be  axes  of  pitch  hypoids  that  will  roll  on 
each  other  in  contact  at  the  directrix. 

Axes  drawn  from  all  points  of  the  line  L 
will  form  a  continuous  surface  called  a  "hy- 
perbolic paraboloid,"  which  will  be  the  locus 
of  all  the  axes  of  a  set  of  hypoids  that  will 
mil  together  collectively  in  contact  at  the 
directrix. 


The  locus  of  axes. 

Fif/.  156. 


98 


Olivier  Skew  Bevel   Gears. 


174. — CYCLOID AL  TEETH  FOR  SKEW  GEARS. 


As  any  number  of  hypoids,  on  axes  in  the 
same  locus  of  axes,  will  roll  together  in 
either  external  or  internal  contact  at  the 
directrix,  it  might  be  supposed  that  a  tooth 
similar  to  the  cycloidal  tooth  for  bevel  and 
spur  gears  might  be  formed  by  an  element  in 
an  auxiliary  hypoid  X,  Fig.  156,  which  rolls 
inside  of  one  and  outside  of  the  other  pitch 
bypoid. 

This  is  such  a  plausible  supposition  that  it 
long  passed  for  the  truth,  not  only  with  its 
inventor,  the  celebrated  Professor  Willis,  but 
with  many  other  prominent  writers,  until 
shown  by  MacCord  to  be  wrong.  It  serves 
to  illustrate  the  confusion  in  which  the  whole 
subject  has  been  and  now  is. 

The  tooth  surfaces  which  Willis  supposed 
to  be  tangent  at  the  generating  element  of 
the  auxiliary  hypoid  really  intersect  at  that 
line,  and  Fig.  157  shows  a  pair  of  such  in- 
tersecting teeth.  The  curves  of  the  figure 
were  drawn  by  an  instrument  made  for  the 


Cycloidal  tooth  Curves 
Fig.  157. 

purpose,  and  are,  therefore,  a  better  proof  of 
the  intersection  of  the  surfaces  than  solid 
teeth  would  be. 

The  cycloidal  tooth  is  examined  at  consid- 
erable length,  and  the  instrumental  proof  of 
its  failure  is  given  in  the  AMERICAN  MACHIN- 
IST for  September  5th,  1889. 


175. — INVOLUTE  TEETH  FOR  SKEW  BEVEL  GEARS. 


Herrmann's  form  of  the  Olivier  spirrloidal 
tooth  is  constructed  with  the  directrix  of 
(172)  as  a  generatrix,  as  follows  : 

Suppose  that  cylinders  are  constructed 
upon  the  gorge  circles  of  a  pair  of  pitch 
hypoids,  Fig.  158,  and  suppose  a  plane  K  to 
be  placed  between  them.  This  plane  will  be 
tangent  to  both  cylinders,  and  will  contain 
the  directrix,  and  if  moved  will  move  the 
^cylinders  as  if  by  friction.  Then  imagine 
ithe  plane  to  move  in  a  direction  normal  to 
the  directrix,  and  it  will  carry  that  directrix 
with  it  as  a  generatrix  always  parallel  to  its 
first  position.  It  will  sweep  up  the  spiraloid 
tooth  surfaces  8^  and  Sa  imperfectly  shown 
by  the  figure,  or  by  Fig.  159,  and  they  will 
be  correct  tooth  surfaces  always  in  tangent 
contact. 

Fig.  159  shows  #,  full  involute  tooth  sur- 
face or  "  spiraloid,"  and  Fig.  160  is  a  full 
Olivier  skew  bevel  gear. 

The  particular  involute  skew  tooth  above 
described  is  not  the  only  possible  form,  but 
it  has  the  least  possible  sliding  action,  and  is, 
therefore,  the  best. 


Involute  tooth  action 
Fig.  158. 

If  the  plane  K  has  a  generatrix  line  at  any 
angles  with  the  axes  of  the  gears,  and  is 
moved  in  a  direction  at  right  angles  with  that 
line,  correct  tooth  surfaces  will  be  swept  up. 
In  fact,  any  two  spiraloids  on  any  two  cylin- 
ders will  work  correctly  with  each  other,  and 
therefore  any  two  spiraloidal  gears  of  the 
same  normal  pitch  will  work  correctly  to- 
gether. 


Beetle's    Gears. 


Olivier  Involute  skew  Bevel  Gear 
Fig.  160. 


176. — HERRMANN'S  LAW. 


Herrmann  gives  a  law,  and  claims  it  to  be 
universal,  to  the  effect  that  the  skew  bevel 
tooth  must  be  swept  up  by  a  straight  line 
generatrix  that  is  always  parallel  to  the  direc- 
trix. He  mentions  the  Olivier  tooth,  and 
claims  that  it  cannot  be  correct,  evidently  not 
understanding  that  Olivier's  theory  clearly 
includes  the  form  he  himself  proposes.  His 
form  of  tooth,  claimed  to  be  the  only  possi- 
ble form,  is  really  only  the  best  form  of  the 
Olivier  tooth. 

We  will  not  undertake  to  state  wherein 
Herrmann's  law  is  incorrect,  but  that  it  is 
wrong  is  clearly  shown  by  the  most  con- 


vincing of  all  proofs,  the  reduction  to  prac- 
tice. Beale,  for  the  Brown  &  Sharpe  Mfg. 
Co.,  has  made  working  Olivier  gears  on  a 
large  scale,  which  are  directly  contrary  to 
Herrmann's  law,  but  which  work  perfectly, 
and  demonstrate  the  truth  of  Olivier's  theory 
in  a  way  that  admits  of  no  question. 

Indeed,  the  closest  possible  scrutiny  of 
Olivier's  theory,  without  the  aid  of  Beale's 
experimental  work,  fails  to  detect  a  flaw  in 
it.  Herrmann's  condemnation  of  it  is  not 
based  on  direct  consideration,  but  simply  on 
the  fact  that  it  does  not  agree  with  his  own 
law. 


177. — BEALE'S  SKEW  BEVEL  GEARS. 


.  Beale's  gears  are  the  same  as  Olivier's  gears 
in  general  theory,  but  the  improvement  in 
practical  form  and  application  is  so  great  that 
they  may  be  considered  a  distinct  invention. 

Fig.  161  is  a  section  through  one  axis,  and 
at  right  angles  to  the  other  axis  of  a  pair  of 
Beale  gears.  Both  surfaces  of  the  teeth  are 
true  Olivier  spiraloids  of  Fig.  159,  and  the 
gears  will  run  in  either  direction.  When 
corrected  for  interference  they  are  reversible, 
like  spur  or  bevel  gears.  The  gorge  cylin- 
ders are  tangent  to  each  other,  and  are  so  cut 
away  inside  as  to  allow  the  teeth  of  the  ma- 
ting gear  to  pass. 

The  Olivier  theory  requires  the  teeth  to 


!  vanish  at  the  gorge,  as  shown  by  the  single 
j  full  tooth  of  Fig.  160,  in  order  to  pass,  while 
i  the  Beale  gear  is  cylindrical  in  form  as  a 
I  whole,  and  passes  the  full  tooth  at  the  gorge, 
with  action  over  its  whole  surface.  The 
;  difference  is  practically  very  great. 

When  in  action  a  pair  of  uncorrected  Beale 
!  gears  must  be  placed  as  shown  by  Figs.  161 
to  163,  and  Fig.  169,  with  one  end  of  each  at 
I  the  gorge,  and  they  will  not  run  together  if 
placed  at  random.      If  either  gear  extends 
beyond  the  gorge  line  there  is  an  interfer- 
ence between  the  involute  spiraloids  which 
is  the  same  in  kind  as  that  between  the  in- 
volute curves  of  common  spur  gear  teeth. 


100 


Beetle's   Gears. 


In  comparison,  the  Beale  gear  is  taken  so 
near  the  gorge  that  it  is  practical  and  service- 
able, having  large  teeth  and  small  obliquity. 


Fig.  161. 


Section  of  Beale's  Gears 


Each  gear  can  drive  in  but  one  direction, 
depending  upon  the  position  of  tbe  gear  and 
the   direction  of  the  spiral,   and  if  turned 
backwards  the  action  is  intermit- 
tent and  practically  useless.     The 
gears  must  be  placed    as  in  Fig. 
162  for  right-hand  spirals,  and  as 
in  Fig.  163  for  left-hand  spirals, 
and  the  direction  of  the  rotation 
is  shown  by  the  arrow  D,  when 
the  gear  bearing  the  arrow  is  the 
driver. 

But,  if  the  direction  is  to  be  re- 
versed, the  gears  can  be  arranged  gorge  cyu. 
as  in  Fig.  162a,  or  as  in  Fig.  163a. 
This  resetting  is  the  same  in  effect 
as  turning  the  gears  half  around, 
except  that  opposite  sides  of  the 
teeth  are  in  contact  in  the  two 
positions  of  the  same  gears. 

If,  however,  the  interfering  parts 
of  the  tooth  surface  are  removed, 
the  gears  will  run  together  per- 
fectly and  in  either  direction  when 
put  together  at  random  as  in  Fig. 
168. 

In  the  cases  shown  by  the  figures, 
the  spirals  make  the  angles  of  forty- 
five  degrees  with  the  shafts,  con- 
trary to  Herrmann's  law,  but  the 
action  will  be  smoother,  and  the 
sliding  of  the  teeth  on  each  other 
will  be  less,  if  Herrmann's  angles 
are  adopted.  These  angles  are  the 
same  as  those  made  by  the  conical 
face  of  common  bevel  gears  of  the 
same  proportion  with  the  axes,  and 
the  best  angles  for  the  two-to-one 
proportion  of  figures  are  those  of 
the  line  X  of  Fig.  162,  making  the 
angles  26°  34'  and  63°  26'  with  the 
axes. 

The  Olivier  gear  of  Fig.  160  is 
perfect  in  theoretical  action,  but  the 
teeth  must  be  taken  so  far  from  the 
gorge  that  the  obliquity  of  the  ac- 
tion is  excessive,  and  the  arc  of 
action  is  so  limited  that  the  teeth* 

must  be  small.  The  sliding  and  wedging  The  working  length  of  each  gear  is  as 
action  is  so  great  that  the  gears  are  practically  determined  by  the  line  L  of  Fig.  161,  and 
useless.  !  the  whole  surface  of  the  tooth  within  that 


1    Fig.  162a. 


Beale  Skew  Bevel  Gears. 

Right  Hand  Spiral  at  45° 


i  Fiy.  163a. 


Left  Hand  Spiral\at  45° 


Approximate   Skew    Teeth. 


101 


limit  will  be  swept  over  "by  the  line  of 
contact.  It  the  length  of  each  gear  is  equal 
to  the  radius  of  the  other  gear  it  will  always 
be  long  enough. 

The  action  between  two  gears  will  be  at 
the  straight,  equidistant,  parallel  lines  a  a, 
Figs.  161  and  162,  in  the  plane  of  action 
tangent  to  both  gorge  cylinders. 

The  shafts  of  a  pair  of  skew  bevel  gears 
should  be  as  near  together  as  possible,  just 
far  enough  apart  to  allow  the  shafts  to  T  ass, 
so  as  to  avoid  the  excessive  sliding  action. 
In  that  case  both  Beale  and  Olivier  gears 
are  practically  useless,  the  former  on  ac 
count  of  the  small  size  of  the  teeth,  and  the 
latter  on  account  of  the  great  obliquity  of 
the  action. 

The  common  bevel  gear  becomes  the  spur 
gear  when  the  shaft  angle  becomes  zero,  but 
the  analogous  transformation  of  the  skew 


bevel  gear  into  a  bevel  gear  by  reducing  the 
gorge  distance  to  zero  is  not  possible. 

The  skew  bevel  gear  becomes  a  spur  gear 
if  we  imagine  the  axes  to  be  brought  parallel 
by  removing  the  gorge  to  an  infinite  distance, 
for  the  spiraloids  on  the  gorge  cylinders  then 
become  involute  surfaces  on  base  cylinders. 
But,  and  it  is  a  curious  circumstance,  when 
the  shafts  are  brought  parallel  by  imagining 
the  shaft  angles  to  become  zero  without 
changing  the  position  of  the  gorge,  the  gorge 
cylinders  become  tangent  and  the  gears  do 
not  become  spur  gears. 

Involute  skew  bevel  gears  do  not  appear 
to  have  any  possible  adjustment  correspond- 
ing to  the  adjustment  of  the  shaft  distance 
of  involute  spur  gears,  or  of  the  shaft  angle 
of  involute  bevel  gears,  (56)  and  (157). 

Beale's  gears  are  fully  described  in  the 
AMERICAN  MACHINIST  of  Aug.  28th,  1890. 


178. — TWISTED    SKEW    TEETH. 


As  no  two  surfaces  of  reference  attached 
to  a  pair  of  revolving  askew  shafts  can  be 
made  to  coincide  with  each  other,  like  the 
planes  of  spur  gears  or  the  spheres  of  bevel 
gears,  the  twisted  or  spiral  tooth  is  impossi- 
ble, for  such  a  tooth  would  not  permit  the 
required  sliding  action. 

But,  if  a  line  is  drawn  upon  one  pitch 
hypoid  of  a  pair,  a  corresponding  line  may 
be  drawn  upon  the  other,  as  if  the  given 


line  could  leave  an  impression.  Therefore 
a  tooth  having  edge  contact  (100)  may  be 
constructed,  provided  the  twist  is  such  that 
one  pair  of  lines  always  crosses  the  directrix. 
These  teeth  are  purely  imaginary,  but  if  the 
edges  are  thick  they  will  have  an  action  upon 
each  other,  at  a  single  point  of  contact,  that 
is  closely  approximate  to  the  theoretical 
action,  and  they  will  serve  the  general  pur- 
pose, if  the  power  carried  is  inconsiderable. 


179. — APPROXIMATE   SKEW   TEETH. 


As  the  true  involute  skew  tooth  is  diffi- 
cult to  construct,  and  in  many  cases  is  of 
small  practical  utility,  and  all  other  proposed 
forms  are  incorrect,  it  follows  that  we  must 
depend  for  practical  purposes  mostly  upon 
some  approximation,  provided  it  is  not  pos- 
sible to  avoid  the  skew  gear  altogether. 

The  blanks  can  be  constructed  by  a  definite 
process  Construct  the  frustra  of  the  pitch 
hypoids  by  the  method  of  (172)  and  Fig.  155. 
Consider  the  end  sections  mn  and  pq  as  ends 
of  a  frustrum  of  a  pitch  cone,  and  on  this 


pitch  cone  construct  the  blank  gear  exactly 
as  for  a  common  bevel  gear. 

Having  constructed  the  blanks,  the  general 
direction  of  the  tooth  is  to  be  marked  upon 
them.  Mount  each  blank  as  in  Fig.  155, 
with  its  axis  parallel  with  a  plane  surface  Z. 
Set  a  surface  gauge  with  its  point  at  the  line 
of  the  directrix  W,  and  with  it  mark  the  po- 
sition of  the  directrix  on  the  pitch  line  at 
each  end  of  the  blank. 

The  tooth  must  then  be  cut  so  that  its 
direction  follows  the  directrix,  and  it  is  to  be 


102 


Substitute   Skew    Trains, 


noticed  that  it  is  not  only  askew  with  the 
axis,  but  that  the  tooth  outline  twists.  The 
appearance  of  the  tooth  on  either  rim,  as 
well  as  upon  any  section  between  the  two 
rims,  is  the  same  as  upon  a  common  bevel 
gear,  symmetrical,  and  not  canted  to  one 
side,  as  is  sometimes  taught. 

The  approximate  tooth  is  very  similar  to 
the  twisted  bevel  tooth,  see  (155)  and  (99), 
with  the  twist  following  a  straight  line  set 
askew  with  the  axis,  and  as  the  line  of  the 
twist  is  not  parallel  with  the  conical  face, 
that  face  should  be  as  short  as  possible. 

The  process  of  cutting  is  not  capable  of 
description,  for  it  depends  upon  personal 
skill  and  judgment.  The  workman  must 
imagine  that  he  sees  the  twisted  cut  in  the 
body  of  the  blank,  and  then  must  persuade 


the  cutter  to  follow  it.  Gear  cutting  ma- 
chines are  seldom  so  made  that  the  cutter 
can  be  turned  while  it  feeds,  and  theretore 
i  it  must  be  set  to  a  medium  path,  and  reset 
|  two  or  three  times  to  get  the  desired  form. 
The  beginner  will  fail  the  first  time,  and 
there  may  be  several  failures.  The  best  pos- 
sible result  can  be  bettered  with  a  file,  after 
running  the  cut  gears  together  to  find  where 
they  interfere. 

In  the  hands  of  a  skillful  workman,  a  pass- 
able approximation  can  be  reached,  and  if  the 
axes  are  very  near  together  compared  with  the 
diameters  of  the  gears,  the  teeth  are  small,  and 
the  face  is  short,  the  result  is  satisfactory. 
In  fact,  when  the  conditions  are  favorable, 
this  approximate  tooth  is  more  serviceable 
than  the  true  tooth. 


Substitute  train. 

Fig.  165. 


180. — SUBSTITUTES  FOR  THE   SKEW  BEVEL 
GEAR. 

When  there  is  a  chance  to  introduce  an 
intermediate  shaft,  the  skew  bevel  gear  can 
be  avoided,  and  it  is  not  only  better,  but 
cheaper  to  avoid  the  objectionable  gear  at 
the  cost  of  the  extra  mechanism. 

Fig.  164  shows  how  to  place  an  inter- 
mediate shaft  and  gears,  when  the  shafts  are 
so  far  apart  that  the  shortest  or  gorge 
distance  can  be  used.  Fig.  165  shows  how 
the  skew  shafts  can  be  connected  by  one 
pair  of  bevel  gears  and  one  pair  of  spur 
gears,  and  that  is  the  best  device  for  general 
purposes. 


Substitute  train. 
Fig.  164. 


Skew  Pin    Gears. 


105 


181.— SKEW  PIN  GEARING. 


Fig.  166  shows  a  pair  of  skew  pin  gears 
commonly  called  face  gears.  They  will  run 
together  with  a  uniform  velocity  ratio  if 
they  are  exactly  alike  and  at  right  angles 
with  shafts  at  a  distance  apart  equal  to  the 
diameter  of  the  pins. 

If  the  gears  are  not  alike  or  not  at  right 
angles,  the  teeth  on  one  may  be  straight 
pins,  but  those  on  the  other  must  be  shaped 
to  correspond. 

Such  gears  are  objectionable  because  they 
have  but  a  single  point  of  contact  for  each 
pair  of  teeth,  at  which  they  slide  on  each 
other  with  great  friction. 

Face  gearing  in  its  various  forms  is  thor- 
oughly examined  in  MacCord's  Kinematics. 
At  the  present  day  they  are  not  in  use,  and 
do  not  deserve  much  study. 


Skew  pin  Ttevel  gears. 

Fig.  206. 


Beale  gears  corrected  for  interference. 
Fig.  168. 


Formation  of  Beale  gear. 
Fig.  167. 


UncorrecUd 
Beale  gears. 

Fig.  169. 


INDEX. 


SECTION. 

Addendum 39 

Arc  of  Action 25 

Axoids 7 

Backlash 41 

Base  Circle 16,  54 

Beale's  Gears 177 

Beale's  Treatise 4 

Begin  at  the  Beginning 3 

Bevel  Gears 6,7,8,  152,  154  to  169 

Box's  Mill  Gearing 4,  51 

Brown  &  Sharpe  Co.'s  Treatise 4 

Cast  Gearing,  Friction 49 

Chart  for  Bevel  Gears 163 

Chordal  Protractor 166 

Clearance 40 

Complete  Tooth 22 

Conic  Pitch  Lines 131 

Conjugator    125 

Consecutive  Action 12 

Construction  by  Points 23,  57 

Cusp 16,  17,  54 

Cutter  Limit 88 

Cutter  Series 45 

Cycloidal  System 31,  76  to  90,  158,  174 

Dedendum 39 

Demonstrations  Avoided 2 

Disputed  Points 9 

Double  Secondary  Action 21  to  78 

Double  Terminal  Action 18 

Edge  Teeth 100 

Efficiency  in  Transmission 49,  73,  112 

Elliptic  Gears 136  to  153 

Elliptic  Gear  Cutting  Machine 150 

Elliptic  Pitch  Lines 131 

Elliptographs 138 

Epicycloidal  Teeth 76 

Equidistant  Series 45 

Extent  of  the  Subject 2 

Fillet 28,  44 

Friction 26 

Friction  of  Approach , 48,  49 

Gear  Cutting  Machines 27,  28,  29, 

94,   102,  105,  122,  125,  150,  159,  168,  169 

Gear  Teeth,  Theory 1  to     34 

"  Spur 35  to     52 

"  Involute 53  to     75 

Cycloidal 76  to     90 

Pin 91  to     97 

Spiral 98  to  110 

Worm Ill  to  125 

"  Irregular 126  to  135 

Elliptic 136  to  153 

Bevel 154  to  169 

SkewBevel. 170  to  181 


SECTION. 

Gears,  Beale's 177 

Bilgram's 159 

Composite 133 

Elliptic  Bevel 152 

Herrmann's 175 

Hindley 122 

Hooke's 98 

Hyperbolic 181 

Lantern t>3 

Mortise 47 

Parabolic 181 

Pin  Bevel 1(50 

"       Skew  Pin 1S1 

Stepped «JH 

Herrmann's  Erroneous  Law 17(1 

Herrmann's  Treatise 4 

Hindley  Worm  Gear 122 

Robbing  Machines 124,  125 

Hobbing  Worm  Gears 114,  115 

Hooke's  Gears 98 

Horse  Power  of  Gears 51,  52 

Hunting  Cog 46 

Hyperbolic  Pitch  Lines 131 

Hyperboloid  of  Revolution 7,  171 

Hypoid 171 

Integrater,  Odontoidal 34 

Interchangeable  Odontoids 14,  22 

Interchangeable  Rack  Tooth 22 

Interference 1 6,  55 

Interference,  Internal 79 

Interference,  Worm 117 

Internal  Contact 15 

Internal  Double  Action 21 

Internal  Gears 64 

Internal  Friction 49 

Involute  System 31,  53  to  75,  157,  175 

Irregular  Pitch  Lines 126  to  128,  133 


Kinematics 

Klein's  Treatise 

Klein's  Odontograph 


85 


Law  of  Tooth  Contact 11,   12 

Limiting  Numbers  of  Teeth 66  to  72,  90 

Limit  Line 16 

Line  of  Action 13 

Literature 4 

Logarithmic  Pitch  Lines 132 

Logarithmic  Spiral 32,  75,   132 

MacCord's  Treatise 4 

Molding  Construction 27 

Mortise  Gear 47 

Multilobes 130 

Natur        Tooth  Action 20 

Normals 11 

Normal  Surfaces 8 


SECTION. 

Obliquity  of  Action 26,  74r,  87 

Octoid  Teeth 31,  159 

Odontics    6 

Odontographs 43,  59,  62,  82,  83 

Odontoid  and  Line  of  Action 33 

Odontoids 12  to  34 

Olivier's  Spiraloidal  Teeth 175 

Parabolic  Pitch  Lines 131 

Parabolic  System 31 

Pin  Tooth  System 91  to  97,  160,  181 

Pitch,  Circular 35,  119 

Pitch  Cylinders 10 

Pitch  Diameters,  Table 35 

Pitch,  Diametral 36,  120 

Pitch  Lines 11 

Pitch  Point 11 

Pitch  Surfaces 7 

Pitch  Table 37 

Planing  Construction 28,  29 

Quick  Return  Motion 149 

Radial  Flank  Teeth 89 

Rankine's  Treatise 4 

Rack  Originator 30,  31 

Retrograde  Action 18 

Reuleaux's  Treatise 4 

Robinson's  Odontograph 86 

Rolled  Curve  Theory 32,  75,  81,  91 

Roller  Teeth 93 


SECTION. 

Secondary  Action 21,  78 

Segmental  System 31 

Sellers'  Experiments 49,  112 

Skew  Bevel  Gears 6,  7,  8,  170  to  181 

Smallest  Pitch  Circle 17 

Speed  of  Point  of  Action   19 

Spiral  Gears 98  to  103 

Spur  Gears 6,  7,  8,  35  to  153 

Stahl  &  Wood's  Treatise 4 

Standard  Teeth 42,  58,  80 

Stepped  Teeth 98 

Strength  of  Teeth 49 

Systems  of  Teeth 31 

Templets 38 

Terminal  Point 18 

Twisted  Teeth .99,  101,  102,   15ft 

Unsymmetrical  Teeth 22 

Variable  Speed 148 

Willis'  Odontograph 84 

Willis'  Treatise 4 

Wooden  Teeth 47 

Worm  Gears ..Ill,  113,  114,  116 

Yale  &  Townes  Experiments 112 


MICHIGAN  BRICK  AND  TILE  MACHINE  CO. 

MORENCI,  MICH.,  Nov.  24th,  1891. 

GEORGE  B.  GRANT.  Dear  Sir  :  —  Two  years  ago  you  sent  me  one  of  your  books  on  Teeth  of 
Gears,  and  I  have  replaced  all  of  the  gears  in  our  brick  machinery  with  new  ones  from  your  in- 
volute odontograph  table.  I  find  that  we  now  have  the  finest  cast  gears  in  the  world.  I  do  not 
understand  why  pattern  makers  don't  catch  on  to  your  book.  It  is  a  sight  to  see  the  gear  pat- 
terns that  are  made  by  some  men  who  are  called  good  pattern  makers. 

O.   S.   STURTEVANT, 

Pattern  Maker  for  M.  B.  &  T.  M.  Co. 


ADVERTISEMENT 


IT'S    NO    USE    TRYING    TO    GET 


ELLIPTIC    GEARS 

OF    ANY    ONE    BUT 

-  \ 

CEO.    B.    GRANT, 

LEXINGTON,  MASS. 
PHILADELPHIA,   PA. 


-n] 
O 
./*/ 

SHi/^/ 


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